// Numbas version: exam_results_page_options
{"name": "Harry's copy of Blathnaid's copy of Differentiate product of trig function and binomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Harry's copy of Blathnaid's copy of Differentiate product of trig function and binomial", "rulesets": {"surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..4)"}, "b": {"name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "s1*random(1..5)"}, "m": {"name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..8)"}, "s1": {"name": "s1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1,-1)"}, "n": {"name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..6)"}}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "extensions": [], "parts": [{"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "gaps": [{"showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "type": "jme", "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "answersimplification": "std", "scripts": {}, "marks": 3, "answer": "{m*b}({a} + {b} * x) ^ {m-1} * sin({n} * x)+{n}*({a} + {b} * x) ^ {m} * cos({n} * x)", "checkvariablenames": false, "showpreview": true}], "showCorrectAnswer": true, "prompt": "<p>$\\simplify[std]{f(x) = ({a} + {b} * x) ^ {m} * sin({n} * x)}$</p>\n<p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>", "variableReplacements": []}], "tags": [], "advice": "\n\t    \n\t    \n\t    <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\t    \n\t    \n\t    \n\t    <p>For this example:</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\simplify[std]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\simplify[std]{v = sin({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * cos({n} * x)}\\]</p>\n\t    \n\t    \n\t    \n\t    <p>Hence on substituting into the product rule above we get:</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\simplify[std]{Diff(f,x,1) = {m*b}({a} + {b} * x) ^ {m-1} * sin({n} * x)+{n}*({a} + {b} * x) ^ {m} * cos({n} * x)}\\]</p>\n\t    \n\t    \n\t", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $ (a+bx) ^ {m} \\sin(nx)$</p>"}, "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/"}]}