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{"name": "Differentiation: Chain Rule 14", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Chain Rule 14", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $\\sin(e^{ax})+b e^{\\cos(cx)}$ using the chain rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the derivative of $y=\\simplify[all,fractionNumbers]{sin(e^({a}x))+{b}e^(cos({c}x))}$.</p>\n<p></p>", "advice": "<p>If we have a function of the form $y=f(g(x))$, sometimes described as a function of a function, to calculate its derivative we need to use the chain rule:</p>\n<p>\\[ \\frac{dy}{dx} = \\frac{du}{dx} \\times \\frac{dy}{du}.\\]</p>\n<p></p>\n<p>This can be split up into steps:</p>\n<ul>\n<li>Let $u=g(x)$;</li>\n<li>Rewrite $y$ in terms of $u$, such that $y=f(u)$;</li>\n<li>Calculate $\\frac{du}{dx}$ and $\\frac{dy}{du}$;</li>\n<li>Write $\\frac{dy}{dx}$ as a product of&nbsp;$\\frac{du}{dx}$ and $\\frac{dy}{du}$;</li>\n<li>Make sure $\\frac{dy}{dx}$ is&nbsp;<em>only</em> in terms of $x$. Ensure any $u$ terms have been replaced using the initial substitution.</li>\n</ul>\n<p>In this case, we have a function of the form $y=f_1(g_1(x))+f_2(g_2(x))$. We want to follow this process for each function, and then add the results together to obtain $\\frac{dy}{dx}$.&nbsp;</p>\n<p>So, for $y=\\simplify[all]{sin(e^({a}x))+{b}e^(cos({c}x))}$, \\[g_1(x)=\\simplify[all]{e^({a}x)}, \\quad g_2(x)=\\simplify[all]{cos({c}x)}.\\]</p>\n<p>If we now set $u_1=g_1(x)$ and $u_2=g_2(x)$, we can rewrite $y$ in terms of $u_1$ and $u_2$ such that $y=f_1(u_1)+f_2(u_2)$:</p>\n<p>\\[y=\\simplify[all]{sin(u_1)+{b}e^(u_2)}.\\]</p>\n<p>Next, we calculate the derivatives $\\frac{du_1}{dx}$, $\\frac{du_2}{dx}$, $\\frac{df_1}{du_1}$ and $\\frac{df_2}{du_2}$:</p>\n<p>\\[ \\begin{split} &amp;\\, \\frac{du_1}{dx}=\\simplify[all]{{a}e^({a}x)}, \\quad &amp;\\,\\frac{df_1}{du_1}=\\simplify[all]{cos(u_1)} \\\\ \\\\&amp;\\,\\frac{du_2}{dx}=\\simplify[all]{-{c}sin({c}x)}, \\quad &amp;\\,\\frac{df_2}{du_2}=\\simplify[all]{{b}e^(u_2)}. \\end{split}\\]</p>\n<p>Plugging these into the extended chain rule:</p>\n<p>\\[ \\begin{split} \\frac{dy}{dx} &amp;= \\frac{du_1}{dx} \\times \\frac{df_1}{du_1} +\\frac{du_2}{dx} \\times \\frac{df_2}{du_2}, \\\\\\\\&amp;=\\simplify[all]{{a}e^({a}x)} \\times\\simplify[all]{cos(u_1) +{-c}sin({c}x)} \\times\\simplify[all]{{b}e^(u_2)} . \\end{split} \\]</p>\n<p>Finally, we need to express $\\frac{dy}{dx}$&nbsp;<em>only&nbsp;</em>in terms of $x$, so we must replace the $u_1$ and $u_2$ terms using the initial substitutions $u_1=\\simplify[all]{e^({a}x)}$ and $u_2=\\simplify[all]{cos({c}x)}$:</p>\n<p>\\[ \\frac{dy}{dx} =\\simplify[all,fractionNumbers]{{a}e^({a}x)cos(e^({a}x))+{-c*b}sin({c}x)e^(cos({c}x))}.\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}e^({a}x)cos(e^({a}x))-{b*c}sin({c}x)e^(cos({c}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": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}