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{"name": "Differentiation: Product Rule 8", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Product Rule 8", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $(ax+b)^n(cx+d)^m$ using the product rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of \\[ \\simplify{y=({a}x+{b})^{n}({c}x+{d})^{m}}. \\]</p>", "advice": "<p>If we have a function of the form $y=u(x)v(x)$, to calculate its derivative we need to use the product rule:</p>\n<p>\\[ \\dfrac{dy}{dx} = u(x) \\times \\dfrac{dv}{dx} + v(x) \\times\\dfrac{du}{dx}.\\]</p>\n<p>This can be split up into steps:</p>\n<ol>\n<li>Identify the functions $u(x)$ and $v(x)$;</li>\n<li>Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;</li>\n<li>Substitute these into the formula for the product rule to obtain an expression for $\\tfrac{dy}{dx}$;</li>\n<li>Simplify $\\tfrac{dy}{dx}$ where possible.</li>\n</ol>\n<p>Following this process, we must first identify $u(x)$ and $v(x)$.</p>\n<p>As \\[ \\simplify{y=({a}x+{b})^{n}({c}x+{d})^{m}}, \\]</p>\n<p>let \\[ u(x) = \\simplify{({a}x+{b})^{n}} \\quad \\text{and} \\quad v(x)=\\simplify{({c}x+{d})^{m}}.\\]</p>\n<p>Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:</p>\n<p>\\[ \\dfrac{du}{dx} = \\simplify{{a*n}({a}x+{b})^{n-1}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{c*m}({c}x+{d})^{m-1}}.\\]</p>\n<p>Substituting these results into the product rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,= \\dfrac{du}{dx}\\times v(x) + u(x) \\times\\dfrac{dv}{dx} \\\\ \\\\&amp;\\,=\\simplify{{a*n}({a}x+{b})^{n-1}} \\times\\simplify{({c}x+{d})^{m}} +\\simplify{({a}x+{b})^{n}} \\times \\simplify{{c*m}({c}x+{d})^{m-1}}.&nbsp; \\end{split}\\]</p>\n<p>Simplifying,</p>\n<p></p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,=\\simplify{{a*n}({a}x+{b})^{n-1}({c}x+{d})^{m}} +\\simplify{({a}x+{b})^{n}{c*m}({c}x+{d})^{m-1}} \\\\ \\\\&amp;\\,= \\simplify{({a}x+{b})^{n-1}({c}x+{d})^{m-1}({n*a}({c}x+{d})+{m*c}({a}x+{b}))}\\\\ \\\\&amp;\\,= \\simplify{({a}x+{b})^{n-1}({c}x+{d})^{m-1}({n*a*c+m*c*a}x+{n*a*d+m*c*b})}\\\\ \\\\&amp;\\,= \\var{simp}\\,\\simplify{ ({a}x+{b})^{n-1}({c}x+{d})^{m-1}({(n*a*c+m*c*a)/simp}x+{(n*a*d+m*c*b)/simp})} .&nbsp; \\end{split}\\]</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..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..6 except [0,a])", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..5 except n)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,b])", "description": "", "templateType": "anything", "can_override": false}, "simp": {"name": "simp", "group": "Ungrouped variables", "definition": "gcd(n*a*c+m*c*a,n*a*d+m*c*b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "simp>1 and simp<20", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "n", "m", "d", "simp"], "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>$\\dfrac{dy}{dx}=$[[0]]&nbsp;</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": "{simp}({a}x+{b})^{n-1}({c}x+{d})^{m-1}({(n*a*c+m*c*a)/simp}x+{(n*a*d+m*c*b)/simp})", "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/"}]}