Quotient rule. (Video)

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

Differentiate the following function $f(x)$ using the quotient rule.

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			Name	Type	Generated Value

a	integer	6
c	integer	1
b	integer	-4
d	integer	9
s2	integer	1
s1	integer	-1
det	integer	58
c1	integer	1

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$\simplify[std]{f(x) = ({a} * x^2+{b})/({c}x^2+{d})}$
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

$g(x)$

$g(x)=\;$ Unnamed gap

Input numbers as fractions or integers and not as decimals.

Click on Show steps for more information. You will not lose any marks by doing so.

You will also find a video in Show steps showing how to apply the quotient rule.

Advice

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:

$\simplify[std]{u = ({a}x^2+{b})}\Rightarrow \simplify[std]{Diff(u,x,1) = {2*a}x}$

$\simplify[std]{v = ({c} * x^2+{d})} \Rightarrow \simplify[std]{Diff(v,x,1) = {2*c}x}$

Hence on substituting into the quotient rule above we get:

$\begin{eqnarray*} \frac{df}{dx}&=&\simplify[std]{({2*a}x({c}x^2+{d})-{2*c}x({a}x^2+{b}))/({c}x^2+{d})^2}\\ &=&\simplify[std]{({2*a*c}x^3+{2*a*d}x-{2*c*a}x^3-{2*c*b}x)/({c}x^2+{d})^2}\\ &=&\simplify[std]{({2*det}x)/({c}x^2+{d})^2} \end{eqnarray*}$
Hence

$g(x)=\simplify[std]{{2*det}x}$

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This question is used in the following exams:

mathcentre: Quotient rule practice by Bill Foster in Bill's workspace.
First Exam by Sarah Jowett in Sarah's workspace.
Bradford: Quotient rule practice by Bill Foster in Bill's workspace.
Maths Support: Quotient rule by Christian Lawson-Perfect in Maths Support Wiki.
Differentiation by Aneela Arshad in Aneela's workspace.
Calculous: 1 by Aneela Arshad in Aneela's workspace.
Differentiation by Bill Foster in Bill's workspace.
Quotient rule by Marlon Arcila in Derivadas.
Blathnaid's copy of Quotient rule by Blathnaid Sheridan in Blathnaid's workspace.

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Bill Foster 55 years, 2 months ago

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Quotient rule. (Video)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`