CC2 Product Rule

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From users who are members of Skills Audits for Maths and Stats :

Will Morgan	said	Ready to use	1 year, 2 months ago
Mash Sheffield	said	Needs to be tested	1 year, 4 months ago

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Will Morgan 1 year ago

Published this.

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Will Morgan 1 year, 2 months ago

Gave some feedback: Ready to use

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Mash Sheffield 1 year, 4 months ago

Gave some feedback: Needs to be tested

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Mash Sheffield 1 year, 4 months ago

Created this as a copy of Differentiation: Product Rule 1a.

Name	Status	Author	Last Modified
Differentiation: Product Rule 1a	Ready to use	Ben McGovern	26/09/2022 11:58
CC2 Product Rule	Ready to use	Mash Sheffield	17/04/2025 09:02

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Question statement

Find the derivative of

$\simplify{y={a}x^{n} sin({b}x)}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	10
b	integer	8
n	integer	2

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Generated value: `integer`

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"Unnamed part" → "Gap 0" - Correct answer

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1. Gap Gap 0 Mathematical expression
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$\dfrac{dy}{dx}=$ Gap 0

Advice

If we have a function of the form $y=u(x)v(x)$ , to calculate its derivative we need to use the product rule:

$\dfrac{dy}{dx} = u(x) \times \dfrac{dv}{dx} + v(x) \times\dfrac{du}{dx}.$

This can be split up into steps:

Identify the functions $u(x)$ and $v(x)$ ;
Calculate their derivatives $\tfrac{du}{dx}$ and $\tfrac{dv}{dx}$ ;
Substitute these into the formula for the product rule to obtain an expression for $\tfrac{dy}{dx}$ ;
Simplify $\tfrac{dy}{dx}$ where possible.

Following this process, we must first identify $u(x)$ and $v(x)$ .

$\simplify{y={a}x^{n} sin({b}x)},$

let

$u(x) = \simplify{{a}x^{n}} \quad \text{and} \quad v(x)=\simplify{sin({b}x)}.$

Next, we need to find the derivatives, $\tfrac{du}{dx}$ and $\tfrac{dv}{dx}$ :

$\dfrac{du}{dx} = \simplify{{a*n}x^{n-1}}\quad \text{and} \quad\dfrac{dv}{dx}=\simplify{{b}cos({b}x)}.$

Substituting these results into the product rule formula we can obtain an expression for $\tfrac{dy}{dx}$ :

$\begin{split} \dfrac{dy}{dx} &\,= \dfrac{du}{dx}\times v(x) + u(x) \times\dfrac{dv}{dx} \\ &\,=\simplify{{a*n}x^{n-1}} \times\simplify{sin({b}x)} +\simplify{{a}x^{n}} \times \simplify{{b}cos({b}x)}. \end{split}$

Simplifying,

$\dfrac{dy}{dx} = \simplify{{n*a}x^{n-1}sin({b}x) + {a*b}x^{n}cos({b}x)}.$

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Will Morgan 1 year ago

Will Morgan 1 year, 2 months ago

Mash Sheffield 1 year, 4 months ago

Mash Sheffield 1 year, 4 months ago

Error in variable testing condition

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Generated value: `integer`

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CC2 Product Rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Will Morgan 1 year ago

Will Morgan 1 year, 2 months ago

Mash Sheffield 1 year, 4 months ago

Mash Sheffield 1 year, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`