// Numbas version: exam_results_page_options
{"name": " Differentiation: Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $f(x) = x^m(a x+b)^n$.</p>"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "name": " Differentiation: Product rule", "parts": [{"type": "gapfill", "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "marks": 0, "steps": [{"type": "information", "variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": [], "prompt": "<p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}}], "gaps": [{"showpreview": true, "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "expectedvariablenames": [], "variableReplacements": [], "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "marks": "2", "checkingtype": "absdiff", "answer": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "vsetrange": [0, 1], "showFeedbackIcon": true, "scripts": {}}], "prompt": "\n      <p>$\\displaystyle \\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "tags": [], "ungrouped_variables": ["a", "s1", "b", "m", "n"], "advice": "\n  \n  \n  <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>\n  \n  \n  \n  <p>For this example:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the product rule above we get:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  ", "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "functions": {}, "extensions": [], "variables": {"n": {"group": "Ungrouped variables", "definition": "random(3..9)", "name": "n", "templateType": "anything", "description": ""}, "m": {"group": "Ungrouped variables", "definition": "random(3..9)", "name": "m", "templateType": "anything", "description": ""}, "b": {"group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "templateType": "anything", "description": ""}, "s1": {"group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "templateType": "anything", "description": ""}, "a": {"group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "templateType": "anything", "description": ""}}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}