Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer

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Xiaodan Leng 5 years, 9 months ago

Created this as a copy of Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer.

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

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Question statement

Differentiate the following function $f(x)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

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

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xxxxxxxxxx
 
random(1..4)

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Generated value: `integer`

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1. Gap Mathematical expression
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$\simplify{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$

You are told that $\simplify{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}$ , for a polynomial $g(x)$ .

You have to find $g(x)$ .

$g(x)=\;$ Unnamed gap

Advice

$f(x)$ is the product of the two functions $\simplify{({a} + {b}*x)^{m}}$ and $\simplify{e ^ ({n} * x)}$ , so we need to use the product rule.

Differentiating the first part, keeping the second half the same, gives the term: $\simplify{{m} *{ b} * ({a} + {b} * x) ^ {m -1}} \times \simplify{e ^ ({n} * x)}$ .

Note that that we needed the chain rule to do this differentiation.

Differentiating the second part, keeping the first half the same, gives the term: $\simplify{{n} * e ^ ({n} * x)} \times \simplify{({a} + {b}x)^{m}}$ .

Again, we needed the chain rule to do this differentiation.

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

$= \simplify{({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}$ , (by doing some factorising)

Hence, $\simplify{g(x) = {m * b + n * a} + {n * b} * x}$ .

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This question is used in the following exam:

Year 2 refresher quiz by Thomas Waters in Thomas's workspace.

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Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer

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Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`