Differentiate the following function $f(x)$ using the quotient rule.
", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "metadata": {"notes": "\n \t\t1/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tAdded information about Show steps. Altered to 0 marks lost rather than 1.
\n \t\tChanged std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.
\n \t\tImproved display in various places.
\n \t\tAdded condition that numbers input as fractions or integers, so added decimal point ot forbidden strings.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "The derivative of $\\displaystyle \\frac{ax+b}{cx^2+d}$ is of the form $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.
"}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": 0, "gaps": [{"checkvariablenames": false, "vsetrange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "marks": 3, "answersimplification": "std", "type": "jme", "showpreview": true, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "notallowed": {"message": "Input all numbers as fractions or integers and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "answer": "{-c*a}x^2+{-2*b*c}x+{a*d}"}], "steps": [{"type": "information", "scripts": {}, "showCorrectAnswer": true, "marks": 0, "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}\\]
\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x^2+{d})}\\]
You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d})^2}\\]
for a polynomial $g(x)$. You are asked to find $g(x)$
$g(x)=\\;$[[0]]
\nInput all numbers as fractions or integers and not as decimals.
\nClick on Show steps for more information. You will not lose any marks by doing so.
\n "}], "type": "question", "showQuestionGroupNames": false, "variable_groups": [], "advice": "\n \n \nThe 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 \n \n \n\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n \n \n \n\\[\\simplify[std]{v = ({c} * x^2+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x}\\]
\n \n \n \nHence on substituting into the quotient rule above we get:
\n \n \n \n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}({c}x^2+{d})-{2*c}x({a}x+{b}))/({c}x^2+{d})^2}\\\\\n \n &=&\\simplify[std]{({a*c}x^2+{a*d}-{2*c*a}x^2-{2*c*b}x)/({c}x^2+{d})^2}\\\\\n \n &=&\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*d})/({c}x^2+{d})^2}\n \n \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*d}}$