Product rule

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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 product rule. The answer will be of the form

$\simplify[std]{({a}x+{b})^{m-1}({c}x+{d})^{n-1}g(x)}$ for a polynomial

$g(x)$ . You have to find

$g(x)$

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			Name	Type	Generated Value

a	integer	7
c	integer	1
b	integer	5
d	integer	-3
s2	integer	-1
s1	integer	1
m	integer	3
n	integer	7
c1	integer	1

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

$g(x)=\;$ 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:

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

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

Hence on substituting into the product rule above we get:

$\simplify[std]{Diff(f,x,1) = {m*a}({a}x+{b}) ^ {m-1} * ({c} * x+{d})^{n}+{n*c}({a}x+{b})^{m} * ({c} * x+{d})^{n-1}=({a}x+{b})^{m-1}({c}x+{d})^{n-1}({m*a*c+n*c*a}x+{m*a*d+n*c*b})}$
on taking out the common terms

$\simplify[std]{({a}x+{b})^{m-1}({c}x+{d})^{n-1}}$ .
So

$\simplify[std]{g(x)= {m*a*c+n*c*a}x+{m*a*d+n*c*b}}$

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Taxonomy: mathcentre

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Product rule

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`