// Numbas version: finer_feedback_settings
{"name": "Chain rule - square root of polynomial,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "variable_groups": [], "functions": {}, "advice": "\n\t  \n\t  \n\t  <p>$\\simplify[std]{f(x) = sqrt({a} * x^{m}+{b})}$<br/>The chain rule says that if $f(x)=g(h(x))$ then<br/>\\[\\simplify[std]{f&#39;(x) = h&#39;(x)g&#39;(h(x))}\\]<br/>One way to find $f&#39;(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.<br/>Then we use the chain rule in the form:<br/>\\[\\frac{df}{dx} = \\frac{du}{dx}\\frac{df(u)}{du}\\]<br/>Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.</p>\n\t  \n\t  \n\t  \n\t  <p>For this example, we let $u=\\simplify[std]{{a} * x^{m}+{b}}$ and we have $f(u)=\\simplify[std]{sqrt(u)=u^{1/2}}$.<br/>This gives<br/>\\[\\begin{eqnarray*}\\frac{du}{dx} &#38;=&#38; \\simplify[std]{{m*a}x ^ {m -1}}\\\\\n\t  \n\t  \\frac{df(u)}{du} &#38;=&#38; \\simplify[std]{{1/2}*u^{-1/2}=1/(2*sqrt(u))} \\end{eqnarray*}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>Hence on substituting into the chain rule above we get:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\begin{eqnarray*}\\frac{df}{dx} &#38;=&#38; \\simplify[std]{{m*a}x ^ {m-1} * (1/(2*sqrt(u)))}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{{m*a}x^{m-1}/(2*sqrt(u))}\\\\\n\t  \n\t  &#38;=&#38; \\simplify[std]{({a*m}x ^ {m-1})/(2*sqrt({a} * x^{m}+{b}))}\n\t  \n\t  \\end{eqnarray*}\\]<br/>on replacing $u$ by $\\simplify[std]{{a}x^{m}+{b}}$.</p>\n\t  \n\t  \n\t", "statement": "<p>Differentiate the following function $f(x)$ using the chain rule.</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["a", "s1", "b", "m"], "tags": [], "extensions": [], "preamble": {"css": "", "js": ""}, "parts": [{"showFeedbackIcon": true, "steps": [{"showFeedbackIcon": true, "variableReplacements": [], "type": "information", "prompt": "\n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t          <p>The chain rule says that if $f(x)=g(h(x))$ then<br/>\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]<br/>One way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.<br/>Then we use the chain rule in the form:<br/>\\[\\frac{df}{dx} = \\frac{du}{dx}\\frac{df}{du}\\]<br/>Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.</p>\n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "scripts": {}, "customMarkingAlgorithm": ""}], "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "gapfill", "stepsPenalty": 0, "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = sqrt({a} * x^{m}+{b})}\\]</p>\n\t\t\t      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n\t\t\t      <p>Click on Show steps for more information. You will not lose any marks by doing so.</p>\n\t\t\t      <p>Input all numbers as fractions or integers and not decimals.</p>\n\t\t\t      <p>&nbsp;</p>\n\t\t\t", "marks": 0, "gaps": [{"checkingAccuracy": 1e-05, "showFeedbackIcon": true, "notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "partialCredit": 0, "showStrings": false, "strings": ["."]}, "expectedVariableNames": [], "vsetRangePoints": 5, "unitTests": [], "vsetRange": [4, 5], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "answer": "({a*m}x ^ {m-1})/(2*sqrt({a} * x^{m}+{b}))", "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "jme", "showPreview": true, "checkVariableNames": false, "marks": 3, "failureRate": 1, "checkingType": "absdiff", "scripts": {}, "customMarkingAlgorithm": ""}], "scripts": {}, "customMarkingAlgorithm": ""}], "name": "Chain rule - square root of polynomial,", "variables": {"m": {"description": "", "name": "m", "definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables"}, "a": {"description": "", "name": "a", "definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables"}, "s1": {"description": "", "name": "s1", "definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables"}}, "metadata": {"description": "\n\t\t<p>Differentiate</p>\n\t\t<p>\\[ \\sqrt{a x^m+b})\\]</p>\n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}