Luis's copy of Chain rule - product of two functions

Error

There was an error loading the page.

Contributors

History

○

Luis Hernandez 6 years, 3 months ago

Created this as a copy of Chain rule - product of two functions.

Name	Status	Author	Last Modified
Chain rule - product of two functions	Ready to use	Newcastle University Mathematics and Statistics	03/11/2020 15:44
Luis's copy of Chain rule - product of two functions	draft	Luis Hernandez	30/11/2018 19:13
Chain rule - product of two functions	Ready to use	Simon Vaughan	20/08/2020 13:00

There are 20 other versions that do you not have access to.

Question statement

Differentiate the following function $f(x)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
s1	integer	-1
b	integer	-3
m	integer	8
n	integer	6

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..9)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

Advice
"Unnamed part" - prompt
"Unnamed part" → "Unnamed gap" - Correct answer

Parts

Gap-fill
1. Gap Mathematical expression
  Add
1. Step Information only

Gap-fill

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

$\simplify[std]{f(x) = x ^ {m} * ({a} * x^2+{b})^{n}}$
The answer is in the form

$\frac{df}{dx}=\simplify[std]{x^{m-1}({a}x^2+{b})^{n-1}*g(x)}$ for a polynomial

$g(x)$ .

You have to find $g(x)$ .

$g(x)=\;$ Unnamed gap

Click on Show steps for 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

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

For this example:

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

$\simplify[std]{v = ({a} * x^2+{b})^{n}} \Rightarrow \simplify[std]{Diff(v,x,1) = {2*n*a}*x * ({a} * x^2+{b})^{n-1}}$

For this last differentiation we used the chain rule.

Hence on substituting into the product rule above we get:

$\begin{eqnarray*}\frac{df}{dx} &=& \simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+x^{m} *{2*n*a}*x* ({a} * x^2+{b})^{n-1}}\\ &=& \simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+{2*n*a}*x^{m+1}* ({a} * x^2+{b})^{n-1}}\\ &=& \simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m}*({a}*x^2+{b})+{2*n*a}x^{2})} \\ &=&\simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m*a+2*a*n}*x^2+{m*b})} \end{eqnarray*}$

Hence $\simplify[std]{g(x)={m*a+2*a*n}*x^2+{m*b}}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Information only

This question is not used in any exams.

Loading...

Error

Luis's copy of Chain rule - product of two functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Luis Hernandez 6 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Luis's copy of Chain rule - product of two functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Luis Hernandez 6 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`