// Numbas version: finer_feedback_settings {"name": "Coursework 3 - Further Differentiation", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": "40", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Differentiation 3 Sine and Cosine functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"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/"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ans22": {"description": "", "definition": "n1[1]", "group": "Ungrouped variables", "templateType": "anything", "name": "ans22"}, "ans31": {"description": "", "definition": "n1[3]*n1[4]*-1", "group": "Ungrouped variables", "templateType": "anything", "name": "ans31"}, "ans12": {"description": "", "definition": "n1[0]", "group": "Ungrouped variables", "templateType": "anything", "name": "ans12"}, "ans11": {"description": "", "definition": "n1[0]", "group": "Ungrouped variables", "templateType": "anything", "name": "ans11"}, "ans41": {"description": "", "definition": "n1[5]", "group": "Ungrouped variables", "templateType": "anything", "name": "ans41"}, "ans21": {"description": "", "definition": "n1[1]*-1", "group": "Ungrouped variables", "templateType": "anything", "name": "ans21"}, "n1": {"description": "", "definition": "shuffle(2..8)[0..6]", "group": "Ungrouped variables", "templateType": "anything", "name": "n1"}, "ans32": {"description": "", "definition": "n1[4]", "group": "Ungrouped variables", "templateType": "anything", "name": "ans32"}}, "tags": ["rebelmaths"], "rulesets": {}, "statement": "

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

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

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

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

Differentiate each of the following:

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

", "variable_groups": [], "preamble": {"js": "", "css": ""}, "functions": {}, "advice": "

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

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

ii)

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

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

iii)

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

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

iv)

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

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

", "ungrouped_variables": ["n1", "ans11", "ans12", "ans21", "ans22", "ans31", "ans32", "ans41"], "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

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

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

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

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

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

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

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

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

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

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

"}, {"marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "checkingtype": "absdiff", "answer": "{ans2}e^({ans2}x)", "type": "jme", "scripts": {}, "checkingaccuracy": 0.001, "checkvariablenames": false, "marks": 1, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "prompt": "

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

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

"}, {"marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "checkingtype": "absdiff", "answer": "{ans3}e^({num2}x)", "type": "jme", "scripts": {}, "checkingaccuracy": 0.001, "checkvariablenames": false, "marks": 1, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "prompt": "

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

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

"}, {"marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "checkingtype": "absdiff", "answer": "1/x", "type": "jme", "scripts": {}, "checkingaccuracy": 0.001, "checkvariablenames": false, "marks": 1, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "prompt": "

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

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

"}, {"marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "checkingtype": "absdiff", "answer": "1/x", "type": "jme", "scripts": {}, "checkingaccuracy": 0.001, "checkvariablenames": false, "marks": 1, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "prompt": "

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

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

"}, {"marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"vsetrangepoints": 5, "checkingtype": "absdiff", "answer": "1/x", "type": "jme", "scripts": {}, "checkingaccuracy": 0.001, "checkvariablenames": false, "marks": 1, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "vsetrange": [0, 1], "showpreview": true, "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "prompt": "

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

$\\frac{dy}{dx} =$ [[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:

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

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

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

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

Differentiate each of the following:

", "functions": {}, "advice": "

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

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

ii)

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

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

iii)

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

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

iv)

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

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

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

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

vi)

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

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

", "tags": ["rebelmaths"], "type": "question"}, {"name": "Differentiation 6 Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}], "functions": {}, "ungrouped_variables": [], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepsPenalty": 0, "prompt": "

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

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

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

Use the following table to help you with the steps:

\n\n\n\n\n\n\n\n\n\n\n\n

$u = $_________________	$v =$_________________
$\\frac{du}{dx} =$_________________	$\\frac{dv}{dx} =$_________________

Find the derivative of y with respect to x given $ y = \\var{n2[0]}\\sqrt x \\ln{\\var{n2[1]}x}$

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

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

Use the following table to help you with the steps:

\n\n\n\n\n\n\n\n\n\n\n\n

$u = $_________________	$v =$_________________
$\\frac{du}{dx} =$_________________	$\\frac{dv}{dx} =$_________________

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

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

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

Use the following table to help you with the steps:

\n\n\n\n\n\n\n\n\n\n\n\n

$u = $_________________	$v =$_________________
$\\frac{du}{dx} =$_________________	$\\frac{dv}{dx} =$_________________

Determine the rate of change of voltage, given $ V = \\var{n4[0]}t \\sin{\\var{n4[1]}t} $ volts when $t = \\var{num4} seconds.$

In formula:

$\\frac{dV}{dt} =$ [[0]]

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

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

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

Use the following table to help you with the steps:

\n\n\n\n\n\n\n\n\n\n\n\n

$u = $_________________	$v =$_________________
$\\frac{du}{dx} =$_________________	$\\frac{dv}{dx} =$_________________

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Remember the Product Rule:

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)}\\]

", "variable_groups": [{"variables": ["n1", "ans11", "ans12"], "name": "q1"}, {"variables": ["n2", "ans21"], "name": "q2"}, {"variables": ["n3", "ans31"], "name": "q3"}, {"variables": ["n4", "num4", "ans41", "ans42"], "name": "q4"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"num4": {"definition": "random(0.2..0.8#0.1)", "templateType": "anything", "group": "q4", "name": "num4", "description": ""}, "ans41": {"definition": "n4[0]*n4[1]", "templateType": "anything", "group": "q4", "name": "ans41", "description": ""}, "ans42": {"definition": "(ans41 * num4 *cos(radians(n4[1]* num4 ))) + (n4[0]* sin(radians( n4[1] * num4)) )", "templateType": "anything", "group": "q4", "name": "ans42", "description": ""}, "ans12": {"definition": "n1[1]-1", "templateType": "anything", "group": "q1", "name": "ans12", "description": ""}, "ans11": {"definition": "n1[0]*n1[1]", "templateType": "anything", "group": "q1", "name": "ans11", "description": ""}, "ans31": {"definition": "n3[0]-1", "templateType": "anything", "group": "q3", "name": "ans31", "description": ""}, "n1": {"definition": "shuffle(3..7)[0..3]", "templateType": "anything", "group": "q1", "name": "n1", "description": ""}, "n2": {"definition": "shuffle(3..7)[0..3]", "templateType": "anything", "group": "q2", "name": "n2", "description": ""}, "n3": {"definition": "shuffle(3..7)[0..3]", "templateType": "anything", "group": "q3", "name": "n3", "description": ""}, "n4": {"definition": "shuffle(3..7)[0..3]", "templateType": "anything", "group": "q4", "name": "n4", "description": ""}, "ans21": {"definition": "n2[0]/2", "templateType": "anything", "group": "q2", "name": "ans21", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Differentiation 7 Quotient Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": 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/"}], "tags": ["rebelmaths"], "preamble": {"js": "", "css": ""}, "statement": "

Remember the Quotient Rule:

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}\\]

", "variables": {"ans15": {"description": "", "definition": "n1[3]*2", "templateType": "anything", "group": "q1", "name": "ans15"}, "ans33": {"description": "", "definition": "((n3[0]*(n3[1]*num31^n3[2] + n3[3]*num31)) - (n3[0]*num31*(ans32*num31^ans31 + n3[3])))/((n3[1]*num31^n3[2] + n3[3]*num31)^2)", "templateType": "anything", "group": "q3", "name": "ans33"}, "ans31": {"description": "", "definition": "n3[2]-1", "templateType": "anything", "group": "q3", "name": "ans31"}, "n1": {"description": "", "definition": "shuffle(3..7)[0..4]", "templateType": "anything", "group": "q1", "name": "n1"}, "n2": {"description": "", "definition": "shuffle(3..7)[0..3]", "templateType": "anything", "group": "q2", "name": "n2"}, "ans14": {"description": "", "definition": "n1[2]^2", "templateType": "anything", "group": "q1", "name": "ans14"}, "ans32": {"description": "", "definition": "n3[1]*n3[2]", "templateType": "anything", "group": "q3", "name": "ans32"}, "ans13": {"description": "", "definition": "n1[2]*n1[3]", "templateType": "anything", "group": "q1", "name": "ans13"}, "num31": {"description": "", "definition": "random(1..3)", "templateType": "anything", "group": "q3", "name": "num31"}, "ans12": {"description": "", "definition": "n1[3]-1", "templateType": "anything", "group": "q1", "name": "ans12"}, "ans11": {"description": "", "definition": "n1[0]*n1[1]", "templateType": "anything", "group": "q1", "name": "ans11"}, "ans21": {"description": "", "definition": "n2[0] * n2[1]", "templateType": "anything", "group": "q2", "name": "ans21"}, "num32": {"description": "", "definition": "random(0.1..0.8#0.05)", "templateType": "anything", "group": "q3", "name": "num32"}, "n3": {"description": "", "definition": "shuffle(3..9)[0..4]", "templateType": "anything", "group": "q3", "name": "n3"}}, "ungrouped_variables": [], "variable_groups": [{"variables": ["n1", "ans11", "ans12", "ans13", "ans14", "ans15"], "name": "q1"}, {"variables": ["n2", "ans21"], "name": "q2"}, {"variables": ["n3", "ans31", "num31", "num32", "ans32", "ans33"], "name": "q3"}], "metadata": {"description": "", "licence": "None specified"}, "parts": [{"stepsPenalty": 0, "extendBaseMarkingAlgorithm": true, "gaps": [{"failureRate": 1, "scripts": {}, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "answer": "((({n1[2]}x^{n1[3]}*{ans11}*cos({n1[1]}x))) - (({n1[0]}*sin({n1[1]}x) * {ans13}x^{ans12})))/({ans14}x^{ans15})", "checkingType": "absdiff", "showPreview": true, "vsetRangePoints": 5, "answerSimplification": "std", "customMarkingAlgorithm": "", "type": "jme", "customName": "", "adaptiveMarkingPenalty": 0, "unitTests": [], "variableReplacements": [], "marks": 1, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "valuegenerators": [{"value": "", "name": "x"}], "checkVariableNames": false, "showFeedbackIcon": true}], "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "unitTests": [], "variableReplacements": [], "steps": [{"extendBaseMarkingAlgorithm": true, "type": "information", "scripts": {}, "showCorrectAnswer": true, "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "unitTests": [], "variableReplacements": [], "marks": 0, "showFeedbackIcon": true, "prompt": "

Use the following table to help you with the steps:

\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]}}} $

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

Use the following table to help you with the steps:

\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} $

$\\frac{dy}{dt} =$ [[0]]

", "customMarkingAlgorithm": ""}, {"stepsPenalty": 0, "extendBaseMarkingAlgorithm": true, "gaps": [{"failureRate": 1, "scripts": {}, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "answer": "(({n3[0]}*({n3[1]}x^{{n3[2]}} + {n3[3]}x)) - ({n3[0]}x*({ans32}x^{ans31} + {n3[3]})))/(({n3[1]}x^{{n3[2]}} + {n3[3]}x)^2)", "checkingType": "absdiff", "showPreview": true, "vsetRangePoints": 5, "answerSimplification": "std", "customMarkingAlgorithm": "", "type": "jme", "customName": "", "adaptiveMarkingPenalty": 0, "unitTests": [], "variableReplacements": [], "marks": 1, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "valuegenerators": [{"value": "", "name": "x"}], "checkVariableNames": false, "showFeedbackIcon": true}, {"mustBeReduced": false, "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "precisionType": "dp", "customMarkingAlgorithm": "", "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "customName": "", "adaptiveMarkingPenalty": 0, "correctAnswerStyle": "plain", "unitTests": [], "variableReplacements": [], "minValue": "{ans33}", "marks": 1, "maxValue": "{ans33}", "precision": "3", "showFeedbackIcon": true, "precisionPartialCredit": 0, "mustBeReducedPC": 0, "correctAnswerFraction": false}], "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "unitTests": [], "variableReplacements": [], "steps": [{"extendBaseMarkingAlgorithm": true, "type": "information", "scripts": {}, "showCorrectAnswer": true, "customName": "", "adaptiveMarkingPenalty": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "unitTests": [], "variableReplacements": [], "marks": 0, "showFeedbackIcon": true, "prompt": "

Use the following table to help you with the steps:

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

In formula:

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

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

$\\frac{dy}{dx} =$ Slope= [[1]]

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

$\\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}} $

ii)

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

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

iii)

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

$\\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} $

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Remember the Chain Rule:

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

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

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

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

ii)

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

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

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

iii)

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

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

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

iv)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Please note you have not completed all questions on this page

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Please note you are about to timeout of the session and your result may not be automatically saved to Minerva. Please remember to save your answers and result at the end.

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Please note you are about to timeout of the session and your result may not be automatically saved to Minerva. Please remember to save your answers and result at the end.

"}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "

This assessment will count towards your module grade. You have unlimited attempts before the deadline. Only your last attempts grade will be saved. Please also save your answers for feedback and proof of result at the end in case it does not automatically save to Minerva.

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