// Numbas version: exam_results_page_options {"name": "Differentiate products of trig, log and exponential terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "b2"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..7)", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "b"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..-1)", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a2"}}, "ungrouped_variables": ["a", "b", "n", "a1", "a2", "b1", "b2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Differentiate products of trig, log and exponential terms", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "
$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$
\n$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a}*sech({a}x+{b})^2", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$
\n$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a2} * tanh({a2} * x + {b2})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$
\n$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "\n \n \nWrite down the derivatives of the following functions $f(x)$ .
\n \n \n \nNote that in order to input the square of a function such as $\\sinh(x)$ you have to input it as $(\\sinh(x))^2$, similarly for the other hyperbolic functions.
\n \n ", "tags": ["Calculus", "chain rule", "checked2015", "cosh", "derivatives of hyperbolic functions", "differential", "differential ", "differentiate", "differentiating hyperbolic functions", "differentiation", "hyperbolic functions", "MAS1601", "product rule", "sinh", "tanh"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "29/06/2012:
\nAdded and edited tags.
\n19/07/2012:
\nAdded description.
\nThere is also the problem of inputting functions of the form $xf(x)$ if $n=1$ or $2$ in the first question. So have reset $n$ to between $3$ and $7$. Otherwise would have to have an instruction here (perhaps depending on value of $n$).
\nChecked calculation.
\n23/07/2012:
\nAdded tags.
\n \nQuestion appears to be working correctly.
\n", "licence": "Creative Commons Attribution 4.0 International", "description": "
Differentiate the following functions: $\\displaystyle x ^ n \\sinh(ax + b),\\;\\tanh(cx+d),\\;\\ln(\\cosh(px+q))$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \nHere is a table of the derivatives of some of the hyperbolic functions:
\n \n \n \n$f(x)$ | $\\displaystyle{\\frac{df}{dx}}$ |
---|---|
$\\sinh(bx)$ | $b\\cosh(bx)$ |
$\\cosh(bx)$ | $b\\sinh(bx)$ |
$\\tanh(bx)$ | $\\simplify{b*sech(bx)^2}$ |
a)
\n \n \n \n$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$
\n \n \n \nUse the product rule to obtain:
\\[\\frac{df}{dx} = \\simplify[std]{{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})}\\]
b)
\n \n \n \n$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$
\n \n \n \nUsing the table above we get:
\\[\\frac{df}{dx} = \\simplify[std]{{a}*sech({a}x+{b})^2}\\]
c)
\n \n \n \n$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$
\n \n \n \nUsing the chain rule we find:
\n \n \n \n\\[\\frac{df}{dx} = \\simplify[std]{{a2} * tanh({a2} * x + {b2})}\\]
\n \n ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}