// Numbas version: exam_results_page_options {"name": "Harry's copy of Differentiation: Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "parts": [{"variableReplacements": [], "marks": 0, "type": "gapfill", "prompt": "\n\t\t\t

\$\\simplify[std]{f(x) = ({a} * x+{b})/({c}*x+{d})}\$

\n\t\t\t

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

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The quotient rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\$

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The quotient rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\$

\n\t \n\t \n\t \n\t

For this example:

\n\t \n\t \n\t \n\t

\$\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\$

\n\t \n\t \n\t \n\t

\$\\simplify[std]{v = ({c} * x+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c}}\$

\n\t \n\t \n\t \n\t

Hence on substituting into the quotient rule above we get:

\n\t \n\t \n\t \n\t

\$\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}({c}x+{d})-{c}({a}x+{b}))/({c}x+{d})^2}\\\\\n\t \n\t &=&\\simplify[std]{({a*c}x+{a*d}-{c*a}x-{c*b})/({c}x+{d})^2}\\\\\n\t \n\t &=&\\simplify[std]{{det}/({c}x+{d})^2}\n\t \n\t \\end{eqnarray*}\$

\n\t \n\t \n\t", "functions": {}, "tags": [], "statement": "

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

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Differentiate $\\displaystyle \\frac{ax+b}{cx+d}$.

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