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{"name": "CF Maths Portfolio - Differentiation 10 - Chain Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["c", "p", "np", "d"], "name": "CF Maths Portfolio - Differentiation 10 - Chain Rule", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p><strong>As before, the solutions here require the use of the 'chain rule'. The process is the same regardless of whether the exterior exponent is negative or even a fraction.</strong></p>\n<p>For example,&nbsp;&nbsp;&nbsp;&nbsp; $y=5(2x^4-7)^{-2}$</p>\n<p>Let $u=2x^4-7$</p>\n<p>This means that: $y=5u^{-2}\\;\\;\\;\\therefore\\;\\;\\;\\frac{dy}{du}=(5\\times-2)u^{-2-1}-10u^{-3}$</p>\n<p>Also: $u=2x^4-7\\;\\;\\;\\therefore\\;\\;\\;\\frac{du}{dx}=(2\\times4)x^{4-1}-(7\\times0)x^{0-1}=8x^3-0=8x^3$</p>\n<p>According to the <em>chain rule:</em></p>\n<p>$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}=-10u^{-3}\\times8x^3=-80x^3u^{-3}$</p>\n<p>Subsituting the original expression back in the place of $u$ gives:</p>\n<p>$\\frac{dy}{dx}=-80x^3(2x^4-7)^{-3}$&nbsp;&nbsp;&nbsp; which may also be written as&nbsp;&nbsp;&nbsp; $\\frac{dy}{dx}=-\\frac{80x^3}{(2x^4-7)^3}$</p>\n<p></p>\n<p>Or else, with a fractional index,&nbsp;&nbsp;&nbsp; $y=\\sqrt[4]{x^2+9}=(x^2+9)^{\\frac{1}{4}}$</p>\n<p>Let $u=x^2+9$</p>\n<p>This means that: $y=u^{\\frac{1}{4}}\\;\\;\\;\\therefore\\;\\;\\;\\frac{dy}{du}=(1\\times\\frac{1}{4})u^{\\frac{1}{4}-1}=\\frac{1}{4}u^{-\\frac{3}{4}}$</p>\n<p>Also: $u=x^2+9\\;\\;\\;\\therefore\\;\\;\\;\\frac{du}{dx}=2x$</p>\n<p>According to the <em>chain rule:</em></p>\n<p>$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}=\\frac{1}{4}u^{-\\frac{3}{4}}\\times2x=\\frac{2}{4}xu^{-\\frac{3}{4}}=\\frac{1}{2}xu^{-\\frac{3}{4}}$</p>\n<p>Subsituting the original expression back in the place of $u$ gives:</p>\n<p>$\\frac{dy}{dx}=\\frac{1}{2}x(x^2+9)^{-\\frac{3}{4}}$&nbsp;&nbsp;&nbsp; which may also be written as&nbsp;&nbsp;&nbsp; $\\dfrac{dy}{dx}=\\dfrac{x}{2(x^2+9)^{\\frac{3}{4}}}$</p>", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "<p>$\\simplify{({c[2]}x^2+{d[2]})^({np[2]})}$</p>", "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{np[2]}*2{c[2]}*x*({c[2]}x^2+{d[2]})^({np[2]}-1)", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>$\\simplify{sqrt({c[5]}x+{d[5]})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "1/2*{c[5]}*({c[5]}x+{d[5]})^(-1/2)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "extensions": [], "statement": "<p><strong>Differentiate the following polynomials with respect to $x$.</strong></p>\n<p>Give&nbsp;your answer in the form $ax^{b}(cx^d+e)^f$, where $f$ is a negative power.</p>\n<p>It is not necessary to include powers of $1$ or terms to the power $0$ in your answer.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"np": {"definition": "repeat(random(-5..-2),10)", "templateType": "anything", "group": "Ungrouped variables", "name": "np", "description": "<p>negative power</p>"}, "p": {"definition": "repeat(random(2..5),10)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": "<p>power</p>"}, "c": {"definition": "repeat(random(2..5),10)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": "<p>coefficient</p>"}, "d": {"definition": "repeat(random(-8..5 except 0),10)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "<p>Using the chain rule with polynomials and negative powers.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}