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

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francisco Glezortiz

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Christian Lawson-Perfect 4 years, 4 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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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

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

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			Name	Type	Generated Value

a	integer	7
s1	integer	-1
b	integer	-9
m	integer	4
n	integer	3

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Differentiate $f(x)$ .

The answer can be written in the form

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

$g(x)$ .

You have to find $g(x)$ .

$g(x)=$ Unnamed gap

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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:

$\begin{align} \frac{\mathrm{d}f}{\mathrm{d}x} &= \simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+x^{m} *{2*n*a}*x* ({a} * x^2+{b})^{n-1}}\\ &= \simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+{2*n*a}*x^{m+1}* ({a} * x^2+{b})^{n-1}}\\ &= \simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m}*({a}*x^2+{b})+{2*n*a}x^{2})} \\ &= \simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m*a+2*a*n}*x^2+{m*b})} \end{align}$

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

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Chain rule - product of two functions

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Christian Lawson-Perfect 4 years, 4 months ago

francisco Glezortiz 4 years, 5 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Chain rule - product of two functions

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Comment

Checkpoint description

Christian Lawson-Perfect 4 years, 4 months ago

francisco Glezortiz 4 years, 5 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`