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 \n \n \n\\[\\simplify[std]{u = ({a}x^2+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {2*a}x}\\]
\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]{({2*a}x({c}x^2+{d})-{2*c}x({a}x^2+{b}))/({c}x^2+{d})^2}\\\\\n \n &=&\\simplify[std]{({2*a*c}x^3+{2*a*d}x-{2*c*a}x^3-{2*c*b}x)/({c}x^2+{d})^2}\\\\\n \n &=&\\simplify[std]{({2*det}x)/({c}x^2+{d})^2}\n \n \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{2*det}x}$
\\[\\simplify[std]{f(x) = ({a} * x^2+{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 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.
\nYou will also find a video in Show steps showing how to apply the quotient rule.
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\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
Differentiate the following function $f(x)$ using the quotient rule.
", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "if(a*d=b*c1,c1+1,c1)", "name": "c"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "d": {"definition": "s2*random(1..9)", "name": "d"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "det": {"definition": "a*d-b*c", "name": "det"}, "c1": {"definition": "random(1..8)", "name": "c1"}}, "metadata": {"notes": "\n \t\t1/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\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 have to be inout as fractions or integers - added decimal point to forbidden strings.
\n \t\t", "description": "The derivative of $\\displaystyle \\frac{ax^2+b}{cx^2+d}$ is $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.
\nContains a video solving a similar quotient rule example. Although does not explicitly find $g(x)$ as asked in the question, but this is obvious.
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