// Numbas version: exam_results_page_options {"name": "Heather's copy of Differentiation : Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "variables": {"b": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b"}, "m": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "name": "m"}, "n": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "name": "n"}, "s1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1"}, "a": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a"}}, "name": "Heather's copy of Differentiation : Product Rule", "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)}\$

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For this example:

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

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

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Hence on substituting into the product rule above we get:

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\$\\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 ", "statement": "

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

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

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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)}\$

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

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

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

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