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.

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			Name	Type	Generated Value

a	integer	8
b	integer	-2
s2	integer	-1
s1	integer	-1
m	integer	4
n	integer	-4

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

$\displaystyle \frac{df}{dx}=\;$ 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:

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

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

Hence on substituting into the product rule above we get:

$\begin{eqnarray*}\frac{df}{dx} &=& \simplify[std]{{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)}\\ &=&\simplify[std]{({b}cos({a}+{b}x)+{n}sin({a}+{b}x))e^({n}x)} \end{eqnarray*}$

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Bill Foster 55 years, 2 months ago

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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

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`