Input all numbers as fractions or integers and not as decimals.
"}, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "showCorrectAnswer": true, "marks": 3, "checkvariablenames": false, "expectedvariablenames": [], "answersimplification": "std", "vsetrangepoints": 5}], "stepsPenalty": 0, "showCorrectAnswer": true, "marks": 0, "scripts": {}, "type": "gapfill", "prompt": "\n\\[\\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 ", "steps": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "type": "information", "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}\\]
Differentiate the following function $f(x)$ using the quotient rule.
", "metadata": {"description": "The derivative of $\\displaystyle \\frac{ax+b}{cx^2+d}$ is of the form $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.
", "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"}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "question_groups": [{"questions": [], "pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered"}], "showQuestionGroupNames": false, "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 $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*d}}$