Chain rule - product of two functions - Numbas at mathcentre.ac.uk

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Simon Vaughan

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4 years, 8 months ago

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Simon Vaughan 4 years, 8 months ago

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Simon Vaughan 4 years, 8 months ago

Gave some feedback: Ready to use

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Simon Vaughan 4 years, 8 months ago

Created this as a copy of Chain rule - product of two functions.

Name	Status	Author	Last Modified
Chain rule - product of two functions	Ready to use	Newcastle University Mathematics and Statistics	03/11/2020 15:44
Luis's copy of Chain rule - product of two functions	draft	Luis Hernandez	30/11/2018 19:13
Chain rule - product of two functions	Ready to use	Simon Vaughan	20/08/2020 13:00

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Question statement

Differentiate the following function $f(x)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
s1	integer	-1
b	integer	-3
m	integer	7
n	integer	4

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random(2..9)

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Generated value: `integer`

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$\simplify[std]{f(x) = x ^ {m} * ({a} * x^2+{b})^{n}}$
The answer is in the form

$\frac{df}{dx}=\simplify[std]{x^{m-1}({a}x^2+{b})^{n-1}*g(x)}$ for a polynomial

$g(x)$ .

You have to find $g(x)$ .

$g(x)=\;$ Unnamed gap

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

Advice

The product rule says that if $u$ and $v$ are functions of $x$ then

$\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}$

For this example:

$\simplify[std]{u = x ^ {m}}\Rightarrow \simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}$

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

For this last differentiation we used the chain rule.

Hence on substituting into the product rule above we get:

$\begin{eqnarray*}\frac{df}{dx} &=& \simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+x^{m} *{2*n*a}*x* ({a} * x^2+{b})^{n-1}}\\ &=& \simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+{2*n*a}*x^{m+1}* ({a} * x^2+{b})^{n-1}}\\ &=& \simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m}*({a}*x^2+{b})+{2*n*a}x^{2})} \\ &=&\simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m*a+2*a*n}*x^2+{m*b})} \end{eqnarray*}$

Hence $\simplify[std]{g(x)={m*a+2*a*n}*x^2+{m*b}}$

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This question is used in the following exams:

Paul's copy of Practice for year 1 by Paul Verheyen in Paul's workspace.
Practice for year 1 by Simon Vaughan in PA1710.
Theme Test 2 by Michael Guy in Michael's workspace.
Melissa's copy of Practice for year 1 by Melissa Heath in Melissa's workspace.
Novry's copy of Practice for year 1 by Novry Erwina in Novry's workspace.

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Chain rule - product of two functions

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Simon Vaughan 4 years, 8 months ago

Simon Vaughan 4 years, 8 months ago

Simon Vaughan 4 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

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Chain rule - product of two functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Simon Vaughan 4 years, 8 months ago

Simon Vaughan 4 years, 8 months ago

Simon Vaughan 4 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`