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+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n \n \n \n\\[\\simplify[std]{v = ({c} * x^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x+{d}}\\]
\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}x+{f})-({2*c}x+{d})({a}x+{b}))/({c}x^2+{d}x+{f})^2}\\\\\n \n &=&\\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}\\\\\n \n &=&\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2}\n \n \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$
\\[\\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]]
\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.
\n ", "gaps": [{"notallowed": {"message": "Input numbers as fractions or integers and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{-c*a}x^2+{-2*b*c}x+{a*f-b*d}", "type": "jme"}], "steps": [{"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.
", "variable_groups": [], "progress": "ready", "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "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"}, "f": {"definition": "random(-9..9)", "name": "f"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "det": {"definition": "a*f-b*d", "name": "det"}, "c1": {"definition": "random(1..8)", "name": "c1"}}, "showQuestionGroupNames": false, "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\n \t\t", "description": "
The derivative of $\\displaystyle \\frac{ax+b}{cx^2+dx+f}$ is $\\displaystyle \\frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.
", "licence": "Creative Commons Attribution 4.0 International"}}, {"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": [], "advice": "\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}\\]
a)
\nFor this example:
\n\\[\\simplify[std]{u = sin({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}cos({a}x)}\\]
\n\\[\\simplify[std]{v = {b}sin({a}x)+{c}cos({a}x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a*b}cos({a}x)+{-a*c}sin({a}x)}\\]
\nHence on substituting into the quotient rule above we get:
\n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}cos({a}x)({b}sin({a}x)+{c}cos({a}x))-sin({a}x)({a*b}cos({a}x)+{-a*c}sin({a}x)))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({a*b} cos({a}x) sin({a}x)+{a*c} cos({a}x)^2-{a*b} sin({a}x)cos({a}x)+{a*c}sin({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({a*c}cos({a}x)^2+{a*c}sin({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({a*c}(cos({a}x)^2+sin({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({a*c})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]
\nHence $a=\\var{a*c}$
\nb)\\[\\simplify[std]{u = cos({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = -{a}sin({a}x)}\\]
\n\\[\\simplify[std]{v = {b}sin({a}x)+{c}cos({a}x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a*b}cos({a}x)+{-a*c}sin({a}x)}\\]
\nHence on substituting into the quotient rule above we get:
\n\\[\\begin{eqnarray*} \\frac{dg}{dx}&=&\\simplify[std]{({-a}sin({a}x)({b}sin({a}x)+{c}cos({a}x))-cos({a}x)({a*b}cos({a}x)+{-a*c}sin({a}x)))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({-a*b}sin({a}x)^2-{a*c} sin({a}x)cos({a}x)-{a*b}cos({a}x)^2+{a*c}sin({a})cos({a}x))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({-a*b}sin({a}x)^2-{a*b}cos({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({-a*b}(sin({a}x)^2+cos({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({-a*b})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]
\nHence $b=\\var{-a*b}$
\nc)
\nWe have that $h(x)=\\simplify[std]{{m}f(x)+{n}g(x)}$
Hence \\[\\begin{eqnarray*}\\frac{dh}{dx} &=& \\simplify[std]{{m}*Diff(f,x,1)+{n}*Diff(f,x,1)}\\\\ &=&\\simplify[std]{{m}*({a*c}/({b}sin({a}x)+{c}cos({a}x))^2)+{n}({-a*b}/({b}sin({a}x)+{c}cos({a}x))^2)}\\\\ &=&\\simplify[std]{(({m}*{a*c})+({n}*{-a*b}))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{{res}/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]
Hence $c=\\var{res}$
\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noleadingMinus"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n\\[\\simplify[std]{f(x) = (sin({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\nYou are given that \\[\\simplify[std]{Diff(f,x,1) = a / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\nfor a number $a$. You have to find $a$.
\n$a=\\;$[[0]]
\n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 3.0, "answer": "{a*c}", "type": "jme"}], "steps": [{"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]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\n \n \n \nYou are given that \\[\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n \n \n \nfor a number $b$. You have to find $b$.
\n \n \n \n$b=\\;$[[0]]
\n \n \n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 3.0, "answer": "{-b*a}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n\\[\\simplify[std]{h(x) = ({m}sin({a}x)+{n}cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\nYou are given that \\[\\simplify[std]{Diff(h,x,1) = c / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\nfor a number $c$. You have to find $c$.
\n$c=\\;$[[0]]
\n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 3.0, "answer": "{res}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "Differentiate the following functions using the quotient rule.
", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "if(b^2=c1^2,c1+1,c1)", "name": "c"}, "b": {"definition": "random(1..9)", "name": "b"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "random(2..9)", "name": "m"}, "n": {"definition": "if(m*c=n1*b,n1+1,n1)", "name": "n"}, "res": {"definition": "m*a*c-n*b*a", "name": "res"}, "n1": {"definition": "s2*random(2..9)", "name": "n1"}, "c1": {"definition": "s1*random(2..8)", "name": "c1"}}, "metadata": {"notes": "\n \t\t\n \t\t
1/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tChanged std rule set to include !noLeadingMinus, so expressions don't change order. Got rid of a redundant ruleset.
\n \t\tImproved display in various places.
\n \t\tChanged to 0 penalty for accessing Show steps in first question.
\n \t\t\n \t\t
\n \t\t", "description": "
Find $\\displaystyle \\frac{d}{dx}\\left(\\frac{m\\sin(ax)+n\\cos(ax)}{b\\sin(ax)+c\\cos(ax)}\\right)$. Three part question.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Quotient rule. (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivative of a quotient", "differentiation", "quotient rule", "steps", "video"], "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^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.
", "gaps": [{"notallowed": {"message": "Input numbers as fractions or integers and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{2*det}x", "type": "jme"}], "steps": [{"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.
", "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.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "showQuestionGroupNames": false, "metadata": {"description": "8 questions on the quotient rule in differentiation.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "exam", "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}], "extensions": [], "custom_part_types": [], "resources": []}