// Numbas version: exam_results_page_options {"metadata": {"description": "

Basic differentiation techniques, including evaluations, brackets/fractions, solving equation, and an application

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A question to test basic differentiation of functions.

", "licence": "None specified"}, "statement": "

Differentiate the following functions with respect to the variable $x$, simplifying your answers where possible.

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$x^\\var{a}-\\sin(\\var{b}x)$

", "answer": "{a}x^{a-1}-{b}cos({b}x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\var{a}e^{-\\var{b}x}+\\frac{\\var{c}}{x}$

", "answer": "-{a}{b}e^(-{b}x)-{c}/x^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\var{d}\\ln(\\var{c}x)-\\var{a*b}\\cos\\left(\\frac{x}{\\var{b}}\\right)$

", "answer": "{d}/x+{a}sin(x/{b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\simplify{{a}*sqrt(x)-{c}*exp({d}*x)+{10*d}*sin({c/10}*x)}$

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$\\simplify{{d}-{b}/x^2+{b-1}x}$

", "answer": "2{b}/x^3+{b-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Differentiate and evaluate", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": ["Calculus", "calculus", "differentiation", "Differentiation"], "metadata": {"description": "

A question to test basic differentiation of functions.

", "licence": "None specified"}, "statement": "

Differentiate the following functions and evaluate them at the given value of $x$.

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If $y=x^\\var{a}-\\cos(\\var{b}x)$, calculate $\\frac{dy}{dx}$ when $x=\\var{x1}$.

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What is the derivative of $x^\\var{a}-\\cos(\\var{b}x)$?

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If $y=\\var{a}e^{\\var{b}x}+\\frac{\\var{c}}{x^2}$, calculate $\\frac{dy}{dx}$ when $x=\\var{x2}$.

What is the derivative of $\\var{a}e^{\\var{b}x}+\\frac{\\var{c}}{x^2}$?

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If $y=\\simplify{{a*b}*sin(x/{b})-{d}*ln({c}*x)}$, calculate $\\frac{dy}{dx}$ when $x=\\var{x3}$.

What is the derivative of $\\simplify{{a*b}*sin(x/{b})-{d}*ln({c}*x)}$?

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If $y=\\simplify{{a}*sqrt(x)+{c}*exp({d}*x)}$, calculate $\\frac{dy}{dx}$ when $x=\\var{x4}$.

What is the derivative of $\\simplify{{a}*sqrt(x)+{c}*exp({d}*x)}$?

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If $y=\\simplify{{d}-{b}/x+{b-1}x}$, calculate $\\frac{dy}{dx}$ when $x=\\var{x5}$.

What is the derivative of $\\simplify{{d}-{b}/x+{b-1}x}$?

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Differentiate the function

\\[h=\\simplify{{a}*t^{n}}(t-\\var{b})(t+\\var{c})\\]

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

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First expand the double bracket: $(t-\\var{b})(t+\\var{c})=$

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Now fully expand $\\simplify{{a}*t^{n}}(t-\\var{b})(t+\\var{c})$:

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Differentiate the function:

\\[r=\\frac{\\simplify{a^{m+1}+{a}*a^{m}+{b}}}{\\simplify{a^{m2}}}\\]

$\\frac{dr}{da}=$ [[0]]

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First separate the fraction into 3 fractions:

\\[\\frac{\\simplify{a^{m+1}+{a}*a^{m}+{b}}}{\\simplify{a^{m2}}}=\\frac{\\simplify{a^{m+1}}}{\\simplify{a^{m2}}}+\\frac{\\simplify{{a}*a^{m}}}{\\simplify{a^{m2}}}+\\frac{\\var{b}}{\\simplify{a^{m2}}}\\]

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Now simplify each fraction:

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Differentiate the following functions which involve brackets or fractions.

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The following questions require you to solve an equation involving derivatives.

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Let $y$ be the following function:

\\[z=\\simplify{a*t^2+b*t+c}\\]

Given that $a=\\var{c}$, $b=\\var{d}$ and $c=\\var{f}$, calculate the value of $t$ such that $\\frac{dz}{dt}=\\var{dz}$.

$t=$ [[0]]

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Substitute the values of the constants $a$, $b$ and $c$ into $z$:

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Now differentiate $z$ to obtain $\\frac{dz}{dt}$:

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Now solve $\\frac{dz}{dt}=\\var{dz}$ by transposing.

Let $y$ be the following function:

\\[y=\\simplify{a*exp(b*x)}\\]

Given that $a=\\var{a}$ and $b=\\var{b}$, calculate the value of $x$ such that $\\frac{dy}{dx}=\\var{dy}$.

$x=$ [[0]]

Substitute the values of the constants $a$ and $b$ into $y$:

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Now differentiate $y$ to obtain $\\frac{dy}{dx}$:

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Now solve $\\frac{dy}{dx}=\\var{dy}$ by transposing and using natural log ($\\ln$).

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The distance, $s$ (in metres), travelled by a body after $t$ (in seconds) is given by the expression

\\[s=\\simplify{{c}+{a}*t+{b}*t^2}\\]

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What is the distance travelled after {t1} {t1s}?

[[0]] metres

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Substitute $t=\\var{t1}$ into the formula for distance.

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Calculate the velocity of the body after {t2} seconds.

[[0]] metres per second

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Differentiate $s$ to obtain a formula for velocity:

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Now substitute $t=\\var{t2}$ into the formula for velocity.

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When does the body come to rest?

[[0]] seconds

The body is at rest when its velocity is zero.

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How far has the body travelled when it comes to rest?

[[0]] metres

From part (c) you know that the body comes to rest after {t3} seconds. Now calculate the distance travelled at this time, i.e. substitute $t=\\var{t3}$ into the formula for distance.

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What is the acceleration of the body?

[[0]] metres per second squared

Differentiate the velocity formula to obtain a formula for acceleration.

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