Differentiate the following function $f(x)$ using the quotient rule.
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"}, "parts": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "stepsPenalty": 0, "customName": "", "prompt": "\n\t\t\t\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x^2+{d}x+{f})}\\]
You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d}x+{f})^2}\\]
for a polynomial $g(x)$. You are asked to find $g(x)$
$g(x)=\\;$[[0]]
\n\t\t\tInput numbers as fractions or integers and not as decimals.
\n\t\t\tClick on Show steps for more information. You will not lose any marks by doing so.
\n\t\t\t", "steps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "customName": "", "prompt": "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}\\]
Input numbers as fractions or integers and not as decimals.
", "strings": ["."]}, "checkingType": "absdiff", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "customMarkingAlgorithm": "", "valuegenerators": [{"value": "", "name": "x"}], "variableReplacements": [], "scripts": {}, "checkVariableNames": false, "failureRate": 1, "marks": 3, "unitTests": [], "customName": "", "answer": "{-c*a}x^2+{-2*b*c}x+{a*f-b*d}", "showCorrectAnswer": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "showPreview": true}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}}], "advice": "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}\\]
For this example:
\n\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n\\[\\simplify[std]{v = ({c} * x^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x+{d}}\\]
\nHence on substituting into the quotient rule above we get:
\n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}({c}x^2+{d}x+{f})-({2*c}x+{d})({a}x+{b}))/({c}x^2+{d}x+{f})^2}\\\\ &=&\\simplify[std]{({a*c}x^2+{a*d}x+{a*f}-{2*c*a}x^2-{a*d+2*c*b}x-{d*b})/({c}x^2+{d}x+{f})^2}\\\\ &=&\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2} \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$