// Numbas version: exam_results_page_options {"name": "Chain Rule 06", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_zWorKza.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_zWorKza.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Chain Rule 06", "tags": [], "metadata": {"description": "Instructional \"drill\" exercise to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

\n

If $y=f(g(x))$  to find $\\frac{dy}{dx}$ , we need to do two things::

\n
    \n
  1. Substitute $u=g(x)$  so that $y=f(u)$
  2. \n
  3. Use the Chain Rule:
  4. \n
\n

\\[ \\large  \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}  \\]

\n

\n
\n

", "advice": "

We are asked to differentiate:

\n

\\[    y=\\ln({\\simplify{{CF}x}+\\sin{(x)})}  \\]

\n

\n

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

\n

\n

Let   $u=\\simplify{{CF}x+sin(x)}$          then          $y=\\ln(u)$

\n

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule and your Table of Derivatives.:

\n

\n

Applying this method gives us:

\n

$\\large \\frac{du}{dx}=\\var{CF}+cos(x)$          and          $ \\large \\frac{dy}{du}= \\frac{1}{u}$

\n

\n

 

\n

We now use the Chain Rule formula:

\n

$ \\large  \\large  \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}   $

\n

Make the appropriate substitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= (\\var{CF}+cos(x)) \\times \\frac{1}{u}$

\n

\n

 

\n

Which simplifies to:

\n

$ \\large  \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{u}$

\n

\n

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem. 

\n

\n

Replace $u$ from our original substitution to give the final answer:

\n

\n

$\\large  \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{\\simplify{{CF}x+sin(x)}}$

\n

\n

\n

\n

\n

 

\n

\n

 

", "rulesets": {}, "extensions": [], "variables": {"CF": {"name": "CF", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["CF"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Differentiate $   y=\\ln({\\var{CF}x+\\sin{(x)})} $

\n

First identify the \"innermost\" function, and substitute $u$:

\n

Let   $u=$[[0]]          Then          $y=$[[1]]

\n

Then:

\n

$  \\large  \\frac{du}{dx}=  $[[2]]          and          $  \\large  \\frac{dy}{du}=  $[[3]]

\n

Now using:

\n

$\\large  \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

\n

\n

$\\large  \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

\n

Which simplifies to:

\n

$\\large  \\frac{dy}{dx}=$[[6]]

\n

\n

Remember that $u$ was a variable that we introduced and not part of the original problem. 

\n

Replace $u$ from our substitution to give the final answer:

\n

$\\large  \\frac{dy}{dx}=$[[7]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{CF}x+sin(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{CF}+cos(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/u", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{CF}+cos(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/u", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({CF}+cos(x))*(1/u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({CF}+cos(x))/({CF}x+sin(x))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}