Find $\\displaystyle \\frac{d}{dx}\\left(\\frac{m\\sin(ax)+n\\cos(ax)}{b\\sin(ax)+c\\cos(ax)}\\right)$. Three part question.
"}, "preamble": {"css": "", "js": ""}, "advice": "\n\tThe 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)
\n\tFor this example:
\n\t\\[\\simplify[std]{u = sin({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}cos({a}x)}\\]
\n\t\\[\\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)}\\]
\n\tHence on substituting into the quotient rule above we get:
\n\t\\[\\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*}\\]
\n\tHence $a=\\var{a*c}$
\n\tb)\\[\\simplify[std]{u = cos({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = -{a}sin({a}x)}\\]
\n\t\\[\\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)}\\]
\n\tHence on substituting into the quotient rule above we get:
\n\t\\[\\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*}\\]
\n\tHence $b=\\var{-a*b}$
\n\tc)
\n\tWe 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 $c=\\var{res}$
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", "parts": [{"customName": "", "type": "gapfill", "customMarkingAlgorithm": "", "steps": [{"customName": "", "type": "information", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "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) = (sin({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\n\t\t\tYou are given that \\[\\simplify[std]{Diff(f,x,1) = a / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n\t\t\tfor a number $a$. You have to find $a$.
\n\t\t\t$a=\\;$[[0]]
\n\t\t\t", "scripts": {}, "useCustomName": false, "stepsPenalty": 0, "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}, {"customName": "", "type": "gapfill", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"customName": "", "type": "jme", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "marks": 3, "showFeedbackIcon": true, "valuegenerators": [], "failureRate": 1, "answer": "{-b*a}", "checkVariableNames": false, "scripts": {}, "useCustomName": false, "showPreview": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}], "sortAnswers": false, "prompt": "\n\t\t\t \n\t\t\t \n\t\t\t\\[\\simplify[std]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\tYou are given that \\[\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\tfor a number $b$. You have to find $b$.
\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t$b=\\;$[[0]]
\n\t\t\t \n\t\t\t \n\t\t\t", "scripts": {}, "useCustomName": false, "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}, {"customName": "", "type": "gapfill", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"customName": "", "type": "jme", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "marks": 3, "showFeedbackIcon": true, "valuegenerators": [], "failureRate": 1, "answer": "{res}", "checkVariableNames": false, "scripts": {}, "useCustomName": false, "showPreview": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}], "sortAnswers": false, "prompt": "\n\t\t\t\\[\\simplify[std]{h(x) = ({m}sin({a}x)+{n}cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\n\t\t\tYou are given that \\[\\simplify[std]{Diff(h,x,1) = c / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n\t\t\tfor a number $c$. You have to find $c$.
\n\t\t\t$c=\\;$[[0]]
\n\t\t\t", "scripts": {}, "useCustomName": false, "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}