Helge's copy of DIfferentiation: product rule

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Helge Münnich

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Helge Münnich 7 years ago

Created this as a copy of DIfferentiation: product rule.

Name	Status	Author	Last Modified
DIfferentiation: product rule	Ready to use	Newcastle University Mathematics and Statistics	20/06/2023 11:26
CINCO SEIS Derivada producto	draft	Marlon Arcila	08/09/2021 00:40
Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer	Ready to use	Lovkush Agarwal	18/06/2018 12:28
Helge's copy of DIfferentiation: product rule	draft	Helge Münnich	28/02/2018 11:26
Helge's copy of DIfferentiation: product rule	Ready to use	Helge Münnich	01/03/2018 01:14
Evan's copy of Differentiation: product and chain rule	draft	Evan Berrett	01/03/2018 15:14
Nick's copy of Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer	draft	Nick Walker	22/06/2018 00:44
Vicki's copy of Nick's copy of Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer	draft	Vicki Fracarossi	28/06/2018 01:21
Praneetha's copy of Nick's copy of Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer	draft	Praneetha Singh	28/06/2018 05:47
Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer [L8 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 10:54
Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer	draft	Xiaodan Leng	11/07/2019 00:11

There are 8 other versions that do you not have access to.

Question statement

Differentiate the following function $f(x)$ using the product rule.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
s1	integer	1
b	integer	1
m	integer	6
n	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..4)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Mathematical expression
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1. Step Mathematical expression

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

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

You are given that

$\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}$

for a polynomial $g(x)$ . You have to find $g(x)$ .

$g(x)=\;$ Unnamed gap

Click on Show steps to get 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

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

For this example:

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

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

Hence on substituting into the product rule above we get:

$\simplify[dPoly]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) = ({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}$

The last step was to take out the common term $\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$ .

Hence

$\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}$ .

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Helge's copy of DIfferentiation: product rule

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Taxonomy: mathcentre

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Helge Münnich 7 years ago

Error in variable testing condition

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Generated value: `integer`

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Error

Helge's copy of DIfferentiation: product rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Helge Münnich 7 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`