// Numbas version: exam_results_page_options {"name": "Adrian's copy of Differentiation: Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

Differentiate the following function $f(x)$ using the product rule.

", "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \n

The product rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\$

\n \n \n \n

For this example:

\n \n \n \n

\$\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\$

\n \n \n \n

\$\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\$

\n \n \n \n

Hence on substituting into the product rule above we get:

\n \n \n \n

\$\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\$

\n \n \n ", "parts": [{"showFeedbackIcon": true, "unitTests": [], "sortAnswers": false, "stepsPenalty": 0, "scripts": {}, "prompt": "\n

$\\displaystyle \\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$

\n

$\\displaystyle \\frac{df}{dx}=\\;$[[0]]

\n

Clicking on Show steps gives you more information, you will not lose any marks by doing so.

\n ", "gaps": [{"showFeedbackIcon": true, "unitTests": [], "checkingAccuracy": 0.001, "checkVariableNames": false, "expectedVariableNames": [], "showPreview": true, "scripts": {}, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "marks": "2", "type": "jme", "failureRate": 1, "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "answerSimplification": "std"}], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "marks": 0, "type": "information", "unitTests": [], "showCorrectAnswer": true, "prompt": "

The product rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\$

", "scripts": {}, "customMarkingAlgorithm": ""}]}], "functions": {}, "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "name": "n", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "description": ""}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "name": "m", "description": ""}}, "metadata": {"description": "

Differentiate $f(x) = x^m(a x+b)^n$.

", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Adrian's copy of Differentiation: Product rule", "tags": [], "extensions": [], "type": "question", "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}