// Numbas version: exam_results_page_options
{"name": "Musa's copy of 3 Chain rule - log of binomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Musa's copy of 3 Chain rule - log of binomial", "tags": [], "metadata": {"description": "<p>Differentiate $\\displaystyle \\ln((ax+b)^{m})$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Differentiate the following function $f(x)$ using the chain rule.</p>", "advice": "<p>$\\simplify[std]{f(x) = ln({a}x+{b}x^{m})}$</p>\n<p>The chain rule applies to $f(x)=g(h(x))$ where</p>\n<p>\\[ g(h) = \\ln(h) {\\rm~and~} \\simplify[std]{h(x) = {a}x+{b}x^{m}}.\\]</p>\n<p>Then we use the chain rule in the form:<br/>\\[\\frac{df}{dx} = \\frac{dh}{dx} \\cdot \\frac{dg}{dh}\\]</p>\n<p>Calculate the derivative of $h(x)$ and $g(h)$: \\[\\frac{dh}{dx} = \\simplify[std]{{a}+{b*m}x^{m-1}}\\]</p>\n<p>\\[\\frac{dg}{dh} = \\frac{1}{h}\\]</p>\n<p>Hence on substituting $h = h(x) = \\simplify[std]{{a}x+{b}x^{m}}$ we finally have</p>\n<p>\\[\\frac{df}{dx} = \\simplify[std]{({a}x+{b*m}x^{m-1})/({a}x+{b}x^{m})} &nbsp;&nbsp; \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "extensions": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "s2", "s1", "m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>\\[\\simplify[std]{f(x) = ln({a}x+{b}x^{m})}\\]</p>\n<p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n<p>Click on Show steps for more information. You will not lose any marks by doing so.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a}+{b*m}x^{m-1})/({a}x+{b}x^{m})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [5, 6], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}