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Differentiation of  polynomials, cos, sin, exp, log functions. Product, quotient and chain rules.

\n

rebelmaths

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Remember the rule:

\n

If $y = ax^n$ then $\\frac{dy}{dx} = anx^{n-1}$

\n

\n

Differentiate each of the following:

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i)

\n

$y = x^{\\var{pow[0]}}$

\n

$\\frac{dy}{dx} = \\var{ans11}x^{\\var{pow[0]}-1} = \\var{ans11}x^{\\var{ans12}}$

\n

ii)

\n

$y = \\var{num[0]}x^{\\var{pow[1]}}$

\n

$\\frac{dy}{dx} = \\var{ans21}x^{(\\var{pow[1]}-1)} = \\var{ans21}x^{\\var{ans22}}$

\n

iii)

\n

$y = \\var{num[1]}x^{\\var{pow[2]}}$

\n

$\\frac{dy}{dx} = \\var{ans31}x^{(\\var{pow[2]}-1)} = \\var{ans31}x^{\\var{ans32}}$

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$y = x^{\\var{pow[0]}}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

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$y = \\var{num[0]}x^{\\var{pow[1]}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num[1]}x^{\\var{pow[2]}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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Remember the rule:

\n

If $y = ax^n$ then $\\frac{dy}{dx} = anx^{n-1}$

\n

\n

Differentiate each of the following:

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i)

\n

$y = x^{\\frac{1}{\\var{p1}}}$

\n

$\\frac{dy}{dx} = \\var{ans11}x^{(\\frac{1}{\\var{p1}}-1)} = \\var{ans11}x^{\\var{ans12}}$

\n

ii)

\n

$y = x^{\\frac{\\var{p2[0]}}{\\var{p2[1]}}}$

\n

$\\frac{dy}{dx} = \\frac{\\var{p2[0]}}{\\var{p2[1]}}x^{(\\frac{\\var{p2[0]}}{\\var{p2[1]}}-1)} = \\var{ans21}x^{\\var{ans22}}$

\n

iii)

\n

$y = \\var{num[0]}x^{\\var{p3}}$

\n

$\\frac{dy}{dx} = \\var{ans31}x^{(\\var{p3}-1)} = \\var{ans31}x^{\\var{ans32}}$

\n

iv)

\n

$y = \\var{num[1]}\\sqrt(x) = \\var{num[1]}x^{\\frac{1}{2}}$

\n

$\\frac{dy}{dx} = \\frac{\\var{num[1]}}{2}x^{(\\frac{1}{2}-1)} = \\var{ans41}x^{\\var{ans42}}$

\n

v)

\n

$y = \\frac{\\var{n51}}{x^{\\var{n52}}} = \\var{n51}x^{-\\var{n52}}$

\n

$\\frac{dy}{dx} = \\var{ans51}x^{(-\\var{n52}-1)} = \\var{ans51}x^{\\var{ans52}}$

\n

vi)

\n

$y = \\sqrt(x^\\var{n6}) = x^{\\frac{\\var{n6}}{2}}$

\n

$\\frac{dy}{dx} = \\frac{\\var{n6}}{2}x^{(\\frac{\\var{n6}}{2}-1)} = \\var{ans61}x^{\\var{ans62}}$

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$y = \\var{num[0]}x^{\\var{p3}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num[1]}\\sqrt x$

\n

$\\frac{dy}{dx} =$ [[0]]

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Hint: What is $\\sqrt x$ as a power of $x$

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$y = \\frac{\\var{n51}}{x^{\\var{n52}}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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Hint: What is $\\frac{1}{x^2}$ as a power of $x$

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$y = \\sin (x) => \\frac{dy}{dx} = \\cos (x)$

\n

$y = \\sin (ax) => \\frac{dy}{dx} = a\\cos (ax)$

\n

$y = \\cos (x) => \\frac{dy}{dx} = -\\sin (x)$

\n

$y = \\cos (ax) => \\frac{dy}{dx} = -a\\sin (ax)$

\n

\n

Differentiate each of the following:

\n

Note: Put the value of the answer of the sine/cosine function in brackets, i.e. write $\\sin2x$ as $\\sin(2x)$

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i)

\n

$y = \\sin \\var{n1[0]}x$

\n

$\\frac{dy}{dx} = \\var{n1[0]}cos(\\var{n1[0]}x)$

\n

ii)

\n

$y = \\cos \\var{n1[1]}x$

\n

$\\frac{dy}{dx} = -\\var{n1[1]}sin(\\var{n1[1]}x)$

\n

iii)

\n

$y = \\var{n1[3]}\\cos \\var{n1[4]}x$

\n

$\\frac{dy}{dx} = (-1 \\times \\var{n1[3]} \\times \\var{n1[4]})sin(\\var{n1[4]}x) = \\var{ans31}sin(\\var{ans32}x)$

\n

iv)

\n

$y = \\var{n1[5]}\\sin x$

\n

$\\frac{dy}{dx} = (\\var{n1[5]} \\times 1)cos(x) = \\var{ans41}cos(x)$

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$y = \\sin (\\var{n1[0]}x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\cos (\\var{n1[1]}x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{n1[3]} \\cos( \\var{n1[4]}x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{n1[5]} \\sin (x)$

\n

$\\frac{dy}{dx} =$ [[0]]

$y = e^{\\var{num[0]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

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$y = e^{\\var{num1}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num[1]}e^{\\var{num2}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\ln \\var{num[2]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\ln \\var{num[3]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\ln \\var{num[4]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

"}], "variables": {"num2": {"definition": "random(-5..-2)", "templateType": "anything", "name": "num2", "group": "Ungrouped variables", "description": ""}, "num": {"definition": "shuffle(2..8)[0..5]", "templateType": "anything", "name": "num", "group": "Ungrouped variables", "description": ""}, "num1": {"definition": "random(0.5..4.5)", "templateType": "anything", "name": "num1", "group": "Ungrouped variables", "description": ""}, "ans1": {"definition": "num[0]", "templateType": "anything", "name": "ans1", "group": "Ungrouped variables", "description": ""}, "ans3": {"definition": "num2*num[1]", "templateType": "anything", "name": "ans3", "group": "Ungrouped variables", "description": ""}, "ans2": {"definition": "num1", "templateType": "anything", "name": "ans2", "group": "Ungrouped variables", "description": ""}}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "metadata": {"licence": "None specified", "description": ""}, "rulesets": {}, "statement": "

Remember the rules:

\n

$y = e^ x => \\frac{dy}{dx} = e^ x$

\n

$y = e^ {ax} => \\frac{dy}{dx} = ae^ {ax}$

\n

$y = \\ln (x) => \\frac{dy}{dx} = \\frac{1}{x}$

\n

$y = \\ln (ax) => \\frac{dy}{dx} = \\frac{1}{x}$

\n

Differentiate each of the following:

i)

\n

$y = e^{\\var{num[0]}x}$

\n

$\\frac{dy}{dx} = \\var{ans1}e^{\\var{num[0]}x}$

\n

ii)

\n

$y = e^{\\var{num1}x}$

\n

$\\frac{dy}{dx} = \\var{ans2}e^{\\var{num1}x}$

\n

iii)

\n

$y = \\var{num[1]}e^{\\var{num2}x}$

\n

$\\frac{dy}{dx} = (\\var{num[1]} \\times \\var{num2})e^{\\var{num2}x} = \\var{ans3}e^{\\var{num2}x}$

\n

iv)

\n

$y = \\ln \\var{num[2]}x$

\n

$\\frac{dy}{dx} = \\frac{1}{x}$

\n

v)

\n

$y = \\ln \\var{num[3]}x$

\n

$\\frac{dy}{dx} = \\frac{1}{x}$

\n

vi)

\n

$y = \\ln \\var{num[4]}x$

\n

$\\frac{dy}{dx} = \\frac{1}{x}$

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$y = \\var{num1[0]}x^\\var{num1[1]} + \\var{num1[2]}x + x^\\var{num1[3]} + \\var{num1[4]}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

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$y = \\var{num2[0]}e^x + \\var{num2[1]}x + \\ln\\var{num2[2]}x + \\var{num2[3]}\\cos (x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = x^\\var{num3[0]} + x^\\var{num3[1]} + \\var{num3[2]}x + \\var{num3[3]}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num5[0]}x^\\var{num5[1]} + \\var{num5[2]}x - \\frac{1}{\\var{num5[3]}x^\\var{num5[4]}} + \\frac{1}{\\sqrt x} - \\var{num5[5]}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

N.B. Put a multipication between the number and the cosine and sine functions in your answer!!

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$y = \\var{num6[0]}\\sin \\var{num6[1]}x + \\var{num6[2]}\\cos \\var{num6[3]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

"}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"num1": {"description": "", "name": "num1", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..5]", "templateType": "anything"}, "ans51": {"description": "", "name": "ans51", "group": "Ungrouped variables", "definition": "num5[0]*num5[1]", "templateType": "anything"}, "ans61": {"description": "", "name": "ans61", "group": "Ungrouped variables", "definition": "num6[0]*num6[1]", "templateType": "anything"}, "num2": {"description": "", "name": "num2", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..4]", "templateType": "anything"}, "ans7": {"description": "", "name": "ans7", "group": "Ungrouped variables", "definition": "num7[0]*num7[1]*-1", "templateType": "anything"}, "ans32": {"description": "", "name": "ans32", "group": "Ungrouped variables", "definition": "num3[1]-1", "templateType": "anything"}, "num7": {"description": "", "name": "num7", "group": "Ungrouped variables", "definition": "shuffle(2..5)[0..2]", "templateType": "anything"}, "ans41": {"description": "", "name": "ans41", "group": "Ungrouped variables", "definition": "num4[0]*num4[1]", "templateType": "anything"}, "ans31": {"description": "", "name": "ans31", "group": "Ungrouped variables", "definition": "num3[0]-1", "templateType": "anything"}, "num3": {"description": "", "name": "num3", "group": "Ungrouped variables", "definition": "shuffle(3..8)[0..4]", "templateType": "anything"}, "num4": {"description": "", "name": "num4", "group": "Ungrouped variables", "definition": "shuffle(3..12)[0..7]", "templateType": "anything"}, "ans63": {"description": "", "name": "ans63", "group": "Ungrouped variables", "definition": "num6[4]*num6[5]", "templateType": "anything"}, "ans43": {"description": "", "name": "ans43", "group": "Ungrouped variables", "definition": "num4[2]*num4[3]", "templateType": "anything"}, "ans13": {"description": "", "name": "ans13", "group": "Ungrouped variables", "definition": "num1[2]", "templateType": "anything"}, "num5": {"description": "", "name": "num5", "group": "Ungrouped variables", "definition": "shuffle(3..10)[0..6]", "templateType": "anything"}, "ans45": {"description": "", "name": "ans45", "group": "Ungrouped variables", "definition": "num4[6]/2", "templateType": "anything"}, "ans11": {"description": "", "name": "ans11", "group": "Ungrouped variables", "definition": "num1[0]*num1[1]", "templateType": "anything"}, "ans52": {"description": "", "name": "ans52", "group": "Ungrouped variables", "definition": "num5[1]-1", "templateType": "anything"}, "ans54": {"description": "", "name": "ans54", "group": "Ungrouped variables", "definition": "(num5[4]*-1)-1", "templateType": "anything"}, "ans44": {"description": "", "name": "ans44", "group": "Ungrouped variables", "definition": "(num4[3]*-1)-1", "templateType": "anything"}, "ans62": {"description": "", "name": "ans62", "group": "Ungrouped variables", "definition": "num6[2]*num6[3]", "templateType": "anything"}, "ans42": {"description": "", "name": "ans42", "group": "Ungrouped variables", "definition": "num4[1]-1", "templateType": "anything"}, "num6": {"description": "", "name": "num6", "group": "Ungrouped variables", "definition": "shuffle(3..12)[0..8]", "templateType": "anything"}, "ans14": {"description": "", "name": "ans14", "group": "Ungrouped variables", "definition": "num1[3]", "templateType": "anything"}, "ans15": {"description": "", "name": "ans15", "group": "Ungrouped variables", "definition": "num1[3]-1", "templateType": "anything"}, "vs": {"description": "", "name": "vs", "group": "Ungrouped variables", "definition": "random('$\\pi$','e')", "templateType": "anything"}, "ans12": {"description": "", "name": "ans12", "group": "Ungrouped variables", "definition": "num1[1]-1", "templateType": "anything"}, "ans53": {"description": "", "name": "ans53", "group": "Ungrouped variables", "definition": "num5[4]/num5[3]", "templateType": "anything"}}, "tags": ["rebelmaths"], "advice": "

i)

\n

$y = \\var{num1[0]}x^\\var{num1[1]} + \\var{num1[2]}x + x^\\var{num1[3]} + \\var{num1[4]}$

\n

$\\frac{dy}{dx} = (\\var{num1[1]} \\times \\var{num1[0]})x^{\\var{num1[1]}-1} + \\var{num1[2]} + (1 \\times \\var{num1[3]})x^{\\var{num1[3]}-1}$

\n

$\\var{ans11}x^(\\var{ans12}) + \\var{ans13}+ \\var{ans14}x^\\var{ans15}$

\n

ii)

\n

$y = \\var{num2[0]}e^x + \\var{num2[1]}x + \\ln\\var{num2[2]}x + \\var{num2[3]}\\cos x$

\n

$\\frac{dy}{dx} = \\var{num2[0]}e^x + \\var{num2[1]} + \\frac{1}{x} - \\var{num2[3]}\\sin x$

\n

iii)

\n

$y = x^\\var{num3[0]} + x^\\var{num3[1]} + \\var{num3[2]}x + \\var{num3[3]}$

\n

$\\frac{dy}{dx} = (1 \\times \\var{num3[0]})x^{\\var{num3[0]}-1} + (1 \\times \\var{num3[1]})x^{\\var{num3[1]}-1} + \\var{num3[2]}$

\n

$\\var{num3[0]}x^\\var{ans31} + \\var{num3[1]}x^\\var{ans32}+ \\var{num3[2]}$

\n

iv)

\n

$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} = (\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + \\var{num4[2]}x^{-\\var{num4[3]}} - \\var{num4[5]} + \\var{num4[6]}x^{\\frac{1}{2}}$

\n

$(\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + (-\\var{num4[3]} \\times \\var{num4[2]})x^{-\\var{num4[3]}-1} - \\var{num4[5]} + (\\frac{1}{2} \\times \\var{num4[6]})x^{\\frac{1}{2}-1}$

\n

$\\var{ans41}x^\\var{ans42} - \\var{ans43}x^\\var{ans44} - \\var{num4[5]} + \\var{ans45}x^{-\\frac{1}{2}}$

\n

v)

\n

$y = \\var{num5[0]}x^\\var{num5[1]} + \\var{num5[2]}x - \\frac{1}{\\var{num5[3]}x^\\var{num5[4]}} + \\frac{1}{\\sqrt x} - \\var{num5[5]}$

\n

$\\frac{dy}{dx} = (\\var{num5[1]} \\times \\var{num5[0]})x^{\\var{num5[1]}-1} + \\var{num5[2]} - \\frac{1}{\\var{num5[3]}}x^{-\\var{num5[4]}} + x^{-\\frac{1}{2}}$

\n

$\\var{ans51}x^\\var{ans52} + \\var{num5[2]} + \\var{ans53}x^\\var{ans54} - \\frac{1}{2}x^{-\\frac{3}{2}}$

\n

vi)

\n

$y = \\var{num6[0]}\\sin \\var{num6[1]}x + \\var{num6[2]}\\cos \\var{num6[3]}x +\\var{num6[4]}e^{\\var{num6[5]}x} + \\var{num6[6]}\\ln\\var{num6[7]}x$

\n

$\\frac{dy}{dx} = (\\var{num6[0]} \\times \\var{num6[1]})\\cos \\var{num6[1]}x - (\\var{num6[2]} \\times \\var{num6[3]})\\sin \\var{num6[3]}x + (\\var{num6[4]} \\times \\var{num6[5]})e^{\\var{num6[5]}x} + \\var{num6[6]} \\times \\frac{1}{x}$

\n

$\\var{ans61}\\cos \\var{num6[1]}x - \\var{ans62}\\sin \\var{num6[3]}x + \\var{ans63}e^{\\var{num6[5]}x} + \\frac{\\var{num6[6]}}{x}$

\n

vii)

\n

$y = \\frac{\\var{num7[0]}}{e^{\\var{num7[1]}x}} + \\var{vs}$

\n

$\\frac{dy}{dx} = \\var{num7[0]}e^{-\\var{num7[1]}x} + \\var{vs}$

\n

$(\\var{num7[0]} \\times -\\var{num7[1]})e^{-\\var{num7[1]}x} + 0$

\n

$\\var{ans7}e^(-\\var{num7[1]}x)$

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Remember the rules from questions 1 to 4:

\n

For each of the following find $\\frac{dy}{dx}$:

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Find the derivative of $y = \\var{n1[0]}x^{\\var{n1[1]}}\\sin{\\var{n1[2]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
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Find the derivative of y with respect to x given  $y = \\var{n2[0]}\\sqrt x \\ln{\\var{n2[1]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({n2[0]}x^(1/2)/x) + ({ans21}x^{-(1/2)} * ln({n2[1]}x))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Differentiate $y = x^{\\var{n3[0]}}\\cos{\\var{n3[1]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
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Determine the rate of change of voltage, given $V = \\var{n4[0]}t \\sin{\\var{n4[1]}t}$ volts when $t = \\var{num4} seconds.$

\n

In formula:

\n

$\\frac{dV}{dt} =$ [[0]]

\n

In units (correct to 2 decimal places!!):

\n

$\\frac{dV}{dt} =$  [[1]] Volts per second

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({ans41}t*cos({n4[1]}t)) + ({n4[0]}* sin({n4[1]}t))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans42}", "strictPrecision": false, "minValue": "{ans42}", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

Remember the Product Rule:

\n

The product rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\$

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Remember the Quotient Rule:

\n

The quotient rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\$

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\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "customMarkingAlgorithm": ""}], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "prompt": "

Find the derivative of $y = \\frac{\\var{n1[0]} \\sin{\\var{n1[1]}x}}{\\var{n1[2]}x^{\\var{n1[3]}}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "customMarkingAlgorithm": ""}], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "prompt": "

Differentiate  $y = \\frac{\\var{n2[0]}e^{\\var{n2[1]}t}}{\\cos \\var{n2[2]}t}$

\n

$\\frac{dy}{dt} =$ [[0]]

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "customMarkingAlgorithm": ""}], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "prompt": "

Find the slope of the curve $y = \\frac{\\var{n3[0]}x}{\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x}$ at the point $(\\var{num31},\\var{num32})$

\n

In formula:

\n

$\\frac{dy}{dx} =$ [[0]]

\n

In units (correct to 3 decimal places!!):

\n

$\\frac{dy}{dx} =$ Slope=  [[1]]

i)

\n

$y = \\frac{\\var{n1[0]} \\sin{\\var{n1[1]}x}}{\\var{n1[2]}x^{\\var{n1[3]}}}$

\n

$\\frac{dy}{dx} = \\frac{((\\var{n1[2]}x^\\var{n1[3]}*\\var{ans11}*\\cos(\\var{n1[1]}x))) - ((\\var{n1[0]}*\\sin(\\var{n1[1]}x) * \\var{ans13}x^\\var{ans12}))}{\\var{ans14}x^\\var{ans15}}$

\n

ii)

\n

$y = \\frac{\\var{n2[0]}e^{\\var{n2[1]}t}}{cos \\var{n2[2]}t}$

\n

$\\frac{dy}{dt} = \\frac{(\\cos(\\var{n2[2]}t) * \\var{ans21} * e^{(\\var{n2[1]}t))} + (\\var{n2[0]} * \\var{n2[2]} * e^{(\\var{n2[1]}t)} * \\sin(\\var{n2[2]}t))}{\\cos(\\var{n2[2]}t)*\\cos(\\var{n2[2]}t)}$

\n

iii)

\n

$y = \\frac{\\var{n3[0]}x}{\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x}$

\n

$\\frac{dy}{dx} = \\frac{(\\var{n3[0]}*(\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x)) - (\\var{n3[0]}x*(\\var{ans32}x^\\var{ans31} + \\var{n3[3]}))}{(\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x)^2}$

\n

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Remember the Chain Rule:

\n

The Chain rule says that if $y$ is a function of $x$, then
$\\frac{dy}{dx} = \\frac{dy}{du}.\\frac{du}{dx}$

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "advice": "

i)

\n

\$\\simplify[std]{y = ({a} * x^{m}+{b})^{n}}\$

\n

$\\frac{dy}{dx} = (\\var{a} \\times \\var{m} \\times \\var{n})x ^ {\\var{m}-1} \\times (\\var{a} \\times x^{\\var{m}}+(\\var{{b}}))^{\\var{n}-1}$

\n

\$\\simplify[std]{{a*m*n}x ^ {m-1} * ({a} * x^{m}+{b})^{n-1}}\$

\n

ii)

\n

\$\\simplify[std]{f(x) = ({a2} x^{m2}+{c2}x^2+{b2})^{n2}}\$

\n

$\\frac{dy}{dt} = \\var{n2}((\\var{a2} \\times \\var{m2})x ^ {\\var{m2}-1}+2 \\times \\var{c2}x) \\times (\\var{a2} \\times x^\\var{m2} + \\var{c2}x^2 + \\var{b2})^{\\var{n2}-1}$

\n

\$\\simplify[std]{{n2}({a2*m2}x ^ {m2-1}+{2*c2}x) * ({a2} * x^{m2}+{c2}x^2+{b2})^{n2-1}}\$

\n

iii)

\n

\$\\simplify[std]{y = sqrt({a3} * x^{m3}+{b3})}\$

\n

$\\frac{dy}{dx} = \\frac{((\\var{a3}*\\var{m3})x ^ {\\var{m3}-1})}{(2*\\sqrt(\\var{a3} * x^\\var{m3}+\\var{b3}))}$

\n

\$\\simplify[std]{({a3*m3}x ^ {m3-1})/(2*sqrt({a3} * x^{m3}+{b3}))}\$

\n

iv)

\n

\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

$\\frac{dy}{dx} = -(\\var{a4}e^{(\\var{a4}x)}+{2*\\var{b4}}x) \\times sin(e^{(\\var{a4}x)} +\\var{b4}x^2+\\var{c4})$

\n

\n

\$\\simplify[std]{-({a4}e^({a4}x)+{2*b4}x)*sin(e^({a4}x) +{b4}x^2+{c4})}\$

\n

v)

\n

\$\\simplify[std]{y = ln(({a5}x+{b5})^{m5})}\$

\n

$\\frac{dy}{dx} = \\frac{({\\var{m5} \\times \\var{a5}})}{(\\var{a5}x+\\var{b5})}$

\n

\$\\simplify[std]{({m5*a5})/({a5}x+{b5})}\$

\n

\n

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\$\\simplify[std]{y = ({a} * x^{m}+{b})^{n}}\$

\n

$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]

\n

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\$\\simplify[std]{y = ({a2} * x^{m2}+{c2}x^2+{b2})^{n2}}\$

\n

$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]

\n

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\$\\simplify[std]{y = sqrt({a3} * x^{m3}+{b3})}\$

\n

$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]

\n

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\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]

\n

", "customName": ""}, {"marks": 0, "unitTests": [], "type": "gapfill", "variableReplacements": [], "useCustomName": false, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "scripts": {}, "gaps": [{"vsetRange": [0, 1], "showCorrectAnswer": true, "variableReplacements": [], "valuegenerators": [{"value": "", "name": "x"}], "checkVariableNames": false, "answer": "({m5*a5})/({a5}x+{b5})", "useCustomName": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "adaptiveMarkingPenalty": 0, "scripts": {}, "customName": "", "marks": 1, "type": "jme", "failureRate": 1, "checkingAccuracy": 0.001, "showPreview": true, "unitTests": [], "vsetRangePoints": 5, "showFeedbackIcon": true, "checkingType": "absdiff"}], "prompt": "

\$\\simplify[std]{f(x) = ln(({a5}x+{b5})^{m5})}\$

\n

$\\displaystyle \\frac{df}{dx}=\\;$[[0]]

\n

", "customName": ""}], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "", "licence": "None specified"}, "variable_groups": [{"name": "q1", "variables": ["a", "s1", "b", "m", "n"]}, {"name": "q2", "variables": ["a2", "c2", "b2", "s22", "s12", "m2", "n2"]}, {"name": "q3", "variables": ["a3", "s13", "b3", "m3"]}, {"name": "q4", "variables": ["a4", "s24", "c4", "b4", "s14"]}, {"name": "q5", "variables": ["a5", "c5", "b5", "s25", "s15", "m5"]}]}]}, {"name": "Group2 Method", "pickingStrategy": "random-subset", "pickQuestions": "2", "questions": [{"name": "Differentiation which method1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "variables": {}, "functions": {}, "advice": "", "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": [], "rulesets": {}, "tags": [], "variable_groups": [], "statement": "

Which rule(s) of differentiation is required to differentiate each of the following functions (tick all that apply).

", "preamble": {"js": "", "css": ""}, "metadata": {"description": "", "licence": "None specified"}, "parts": [{"distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of two or more functions."], "shuffleChoices": false, "minMarks": 0, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "displayColumns": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "

$y=e^x\\cos(x)$

", "unitTests": [], "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "marks": 0, "type": "1_n_2", "matrix": ["1", 0, 0], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup"}, {"distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of tow or more functions."], "shuffleChoices": false, "minMarks": 0, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "displayColumns": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "

$y=\\frac{e^x}{\\cos(x)+2}$

", "unitTests": [], "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "marks": 0, "type": "1_n_2", "matrix": ["0", "1", 0], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup"}, {"distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of tow or more functions."], "shuffleChoices": false, "minMarks": 0, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "scripts": {}, "displayColumns": 0, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "

$y=\\cos(e^x)$

", "unitTests": [], "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "marks": 0, "type": "1_n_2", "matrix": ["0", "0", "1"], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup"}], "type": "question"}, {"name": "Differentiation which method2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "metadata": {"licence": "None specified", "description": ""}, "variables": {}, "parts": [{"minMarks": 0, "unitTests": [], "type": "1_n_2", "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "displayType": "radiogroup", "marks": 0, "matrix": ["0", "1", 0], "distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of two or more functions."], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "prompt": "

$y=\\frac{\\sin(x)}{x^4+1}$

", "variableReplacements": [], "maxMarks": 0, "shuffleChoices": false}, {"minMarks": 0, "unitTests": [], "type": "1_n_2", "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "displayType": "radiogroup", "marks": 0, "matrix": ["0", "0", "1"], "distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of two or more functions."], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "prompt": "

$y=(x^4+5x)^7$

", "variableReplacements": [], "maxMarks": 0, "shuffleChoices": false}, {"minMarks": 0, "unitTests": [], "type": "1_n_2", "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "displayType": "radiogroup", "marks": 0, "matrix": ["1", "0", "0"], "distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of two or more functions."], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "prompt": "

$y=x^2e^x$

", "variableReplacements": [], "maxMarks": 0, "shuffleChoices": false}], "ungrouped_variables": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "advice": "", "rulesets": {}, "tags": [], "statement": "

Which rule(s) of differentiation is required to differentiate each of the following functions (tick all that apply).

", "type": "question"}, {"name": "Differentiation which method3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "tags": [], "ungrouped_variables": [], "parts": [{"distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of two or more functions."], "prompt": "

$y=x^4\\sin(x)$

", "maxMarks": 0, "unitTests": [], "marks": 0, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "scripts": {}, "displayType": "radiogroup", "extendBaseMarkingAlgorithm": true, "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "displayColumns": 0, "matrix": ["1", "0", 0], "minMarks": 0, "showFeedbackIcon": true, "shuffleChoices": false, "type": "1_n_2"}, {"distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of tow or more functions."], "prompt": "

$y=(x^4+5x)^2$

", "maxMarks": 0, "unitTests": [], "marks": 0, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "scripts": {}, "displayType": "radiogroup", "extendBaseMarkingAlgorithm": true, "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "displayColumns": 0, "matrix": ["0", "0", "1"], "minMarks": 0, "showFeedbackIcon": true, "shuffleChoices": false, "type": "1_n_2"}, {"distractors": ["This rule is used when you are differentiating a function which is made up of one function multiplied by another function.", "This rule is used when you are differentiating a function which is made up of one function divided by another function.", "This rule is used when you are differentiating a function which is the composition of tow or more functions."], "prompt": "

$y=\\frac{x^2}{e^x+4}$

", "maxMarks": 0, "unitTests": [], "marks": 0, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "scripts": {}, "displayType": "radiogroup", "extendBaseMarkingAlgorithm": true, "choices": ["Product Rule", "Quotient Rule", "Chain Rule"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "displayColumns": 0, "matrix": ["0", "1", "0"], "minMarks": 0, "showFeedbackIcon": true, "shuffleChoices": false, "type": "1_n_2"}], "rulesets": {}, "variable_groups": [], "variables": {}, "functions": {}, "statement": "

Which rule(s) of differentiation is required to differentiate each of the following functions (tick all that apply).

", "metadata": {"licence": "None specified", "description": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "advice": "", "type": "question"}]}, {"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Differentiation 9 All Types", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "parts": [{"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{ans11}x^{ans12}"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$y = \\var{num1}x^{\\var{pow1}}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{ans21}x^{ans22}"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$y = \\frac{\\var{n21}}{x^{\\var{n22}}}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{ans31}sin({ans32}x)"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$y = \\var{n3[0]} \\cos \\var{n3[1]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{ans4}e^({n42}x)"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$y = \\var{n41}e^{\\var{n42}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{ans41}x^{ans42} - {ans43}x^{ans44} - {num4[5]} + {ans45}x^{-(1/2)}"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "({n6[0]}x^(1/2)/x) + ({ans61}x^{-(1/2)} * ln({n6[1]}x))"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find the derivative of y with respect to x given  $y = \\var{n6[0]}\\sqrt x \\ln{\\var{n6[1]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "t"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "((cos({n7[2]}t) * {ans71} * e^({n7[1]}t)) + ({n7[0]} * {n7[2]} * e^({n7[1]}t) * sin({n7[2]}t)))/(cos({n7[2]}t)*cos({n7[2]}t))"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Differentiate  $y = \\frac{\\var{n7[0]}e^{\\var{n7[1]}t}}{\\cos \\var{n7[2]}t}$

\n

$\\frac{dy}{dt} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "-({a4}e^({a4}x)+{2*b4}x)*sin(e^({a4}x) +{b4}x^2+{c4})"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}], "ungrouped_variables": ["num1", "pow1", "ans11", "ans12"], "tags": ["rebelmaths"], "advice": "

i)

\n

$y = \\var{num1}x^{\\var{pow1}}$

\n

$\\frac{dy}{dx} = \\var{ans11}x^{(\\var{pow1}-1)} = \\var{ans11}x^{\\var{ans12}}$

\n

ii)

\n

$y = \\frac{\\var{n21}}{x^{\\var{n22}}} = \\var{n21}x^{-\\var{n22}}$

\n

$\\frac{dy}{dx} = \\var{ans21}x^{(-\\var{n22}-1)} = \\var{ans21}x^{\\var{ans22}}$

\n

iii)

\n

$y = \\var{n3[0]}\\cos \\var{n3[1]}x$

\n

$\\frac{dy}{dx} = (-1 \\times \\var{n3[0]} \\times \\var{n3[1]})sin(\\var{n3[1]}x) = \\var{ans31}sin(\\var{ans32}x)$

\n

iv)

\n

$y = \\var{n41}e^{\\var{n42}x}$

\n

$\\frac{dy}{dx} = (\\var{n41} \\times \\var{n42})e^{\\var{n42}x} = \\var{ans3}e^{\\var{n42}x}$

\n

v)

\n

$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} = (\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + \\var{num4[2]}x^{-\\var{num4[3]}} - \\var{num4[5]} + \\var{num4[6]}x^{\\frac{1}{2}}$

\n

$(\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + (-\\var{num4[3]} \\times \\var{num4[2]})x^{-\\var{num4[3]}-1} - \\var{num4[5]} + (\\frac{1}{2} \\times \\var{num4[6]})x^{\\frac{1}{2}-1}$

\n

$\\var{ans41}x^\\var{ans42} - \\var{ans43}x^\\var{ans44} - \\var{num4[5]} + \\var{ans45}x^{-\\frac{1}{2}}$

\n

vi)

\n

$y = \\var{n6[0]}\\sqrt x \\ln{\\var{n6[1]}x}$

\n

$\\frac{dy}{dx} = (\\frac{\\var{n6[0]}x^{\\frac{1}{2}}}{x}) + (\\var{ans61}x^{-\\frac{1}{2}} \\times ln(\\var{n6[1]}x))$

\n

vii)

\n

$y = \\frac{\\var{n7[0]}e^{\\var{n7[1]}t}}{cos \\var{n7[2]}t}$

\n

$\\frac{dy}{dt} = \\frac{(\\cos(\\var{n7[2]}t) * \\var{ans71} * e^{(\\var{n7[1]}t))} + (\\var{n7[0]} * \\var{n7[2]} * e^{(\\var{n7[1]}t)} * \\sin(\\var{n7[2]}t))}{\\cos(\\var{n7[2]}t)*\\cos(\\var{n7[2]}t)}$

\n

viii)

\n

\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

$\\frac{dy}{dx} = -(\\var{a4}e^{(\\var{a4}x)}+{2*\\var{b4}}x) \\times sin(e^{(\\var{a4}x)} +\\var{b4}x^2+\\var{c4})$

\n

\n

\$\\simplify[std]{-({a4}e^({a4}x)+{2*b4}x)*sin(e^({a4}x) +{b4}x^2+{c4})}\$

", "statement": "

Remember all the rules from the previous questions and use them to solve the following:

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"anything"}, "ans22": {"description": "", "definition": "(n22*-1)-1", "name": "ans22", "group": "q2", "templateType": "anything"}, "pow1": {"description": "", "definition": "random(3..7)", "name": "pow1", "group": "Ungrouped variables", "templateType": "anything"}, "n41": {"description": "", "definition": "random(2..8)", "name": "n41", "group": "q4", "templateType": "anything"}, "num4": {"description": "", "definition": "shuffle(3..12)[0..7]", "name": "num4", "group": "q5", "templateType": "anything"}, "ans12": {"description": "", "definition": "pow1-1", "name": "ans12", "group": "Ungrouped variables", "templateType": "anything"}, "ans44": {"description": "", "definition": "(num4[3]*-1)-1", "name": "ans44", "group": "q5", "templateType": "anything"}, "ans11": {"description": "", "definition": "pow1 * num1", "name": "ans11", "group": "Ungrouped variables", "templateType": "anything"}, "ans61": {"description": "", "definition": "n6[0]/2", "name": "ans61", "group": "q6", "templateType": "anything"}, "ans32": {"description": "", "definition": "n3[1]", "name": "ans32", "group": "q3", "templateType": "anything"}, "ans4": {"description": "", "definition": "n42*n41", "name": "ans4", "group": "q4", "templateType": "anything"}, "n21": {"description": "", "definition": "random(3..6)", "name": "n21", "group": "q2", "templateType": "anything"}, "ans45": {"description": "", "definition": "num4[6]/2", "name": "ans45", "group": "q5", "templateType": "anything"}, "n42": {"description": "", "definition": "random(-5..-2)", "name": "n42", "group": "q4", "templateType": "anything"}, "s14": {"description": "", "definition": "random(1,-1)", "name": "s14", "group": "q8", "templateType": "anything"}, "s24": {"description": "", "definition": "random(1,-1)", "name": "s24", "group": "q8", "templateType": "anything"}, "ans21": {"description": "", "definition": "n21*n22*-1", "name": "ans21", "group": "q2", "templateType": "anything"}, "n7": {"description": "", "definition": "shuffle(3..7)[0..3]", "name": "n7", "group": "q7", "templateType": "anything"}, "n22": {"description": "", "definition": "random(3..6 except n21)", "name": "n22", "group": "q2", "templateType": "anything"}, "ans31": {"description": "", "definition": "n3[0]*n3[1]*-1", "name": "ans31", "group": "q3", "templateType": "anything"}, "n6": {"description": "", "definition": "shuffle(3..7)[0..3]", "name": "n6", "group": "q6", "templateType": "anything"}, "a4": {"description": "", "definition": "random(2..9)", "name": "a4", "group": "q8", "templateType": "anything"}}, "variable_groups": [{"name": "q2", "variables": ["n21", "n22", "ans21", "ans22"]}, {"name": "q3", "variables": ["n3", "ans31", "ans32"]}, {"name": "q4", "variables": ["n41", "n42", "ans4"]}, {"name": "q5", "variables": ["num4", "ans41", "ans42", "ans43", "ans44", "ans45"]}, {"name": "q6", "variables": ["n6", "ans61"]}, {"name": "q7", "variables": ["n7", "ans71"]}, {"name": "q8", "variables": ["a4", "s24", "c4", "b4", "s14"]}], "metadata": {"description": "", "licence": "None specified"}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}}]}], "duration": 0, "showstudentname": true, "feedback": {"showactualmark": true, "showtotalmark": true, "allowrevealanswer": true, "intro": "", "feedbackmessages": [], "showanswerstate": true, "advicethreshold": 0}, "showQuestionGroupNames": false, "timing": {"timedwarning": {"message": "", "action": "none"}, "allowPause": true, "timeout": {"message": "", "action": "none"}}, "type": "exam", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Gareth Woods", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/978/"}, {"name": "Johnny Yi", "profile_url": 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