Product Rule 05

Error

There was an error loading the page.

Contributors

Michael Proudman

Feedback

From users who are members of .Differential Calculus :

Michael Proudman

said

Needs to be tested

4 years, 10 months ago

History

○

Michael Proudman 4 years, 10 months ago

Gave some feedback: Needs to be tested

○

Michael Proudman 4 years, 10 months ago

Published this.

○

Michael Proudman 4 years, 10 months ago

Created this.

Name	Status	Author	Last Modified
Product Rule 05	Needs to be tested	Michael Proudman	07/12/2021 08:32
Product Rule 05 (non scaffold)	Needs to be tested	Michael Proudman	07/12/2021 08:32

There is one other version that you do not have access to.

Question statement

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \times v$ then:

$\frac{dy}{dx}=u \frac{dv}{dx} + v \frac{du}{dx}$

Give any introductory information the student needs.

			Name	Type	Generated Value

aCF	integer	6
aP	integer	4
eP	integer	6

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..6)

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" → "Gap 0." - Correct answer
"Unnamed part" → "Gap 2." - Correct answer
"Unnamed part" → "Gap 4." - Correct answer
"Unnamed part" → "Gap 7." - Correct answer
"Unnamed part" → "Gap 8." - Correct answer

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
3. Gap Gap 2. Mathematical expression
  Add
4. Gap Gap 3. Mathematical expression
  Add
5. Gap Gap 4. Mathematical expression
  Add
6. Gap Gap 5. Mathematical expression
  Add
7. Gap Gap 6. Mathematical expression
  Add
8. Gap Gap 7. Mathematical expression
  Add
9. Gap Gap 8. Mathematical expression
  Add

Gap-fill

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

Differentiate $y=\var{aCF}x^{\var{aP}} e^{\var{eP}x}$

First identify the two functions $u$ and $v$ :

$u=$ Gap 0. $v=$ Gap 1.

Now differentiate each one:

$\large \frac{du}{dx}=$ Gap 2. $\large\frac{dv}{dx}=$ Gap 3.

Substitute each component into the formula in the correct place:

$\large \frac{dy}{dx}=$ Gap 4. $\large \times$ Gap 5. $\large +$ Gap 6. $\large \times$ Gap 7.

Finally tidy this up to give your final answer:

$\large \frac{dy}{dx}=$ Gap 8.

Advice

We are asked to differentiate:

$y=\var{aCF}x^{\var{aP}} e^{\var{eP}x}$

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$\large u=\var{aCF}x^{\var{aP}}$ $\large v=e^{\var{eP}x}$

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we can use the Power Rule and for $v$ your Table of Derivatives:

This gives us:

$\large \frac{du}{dx}=\simplify{ {aP}*{aCF}*x^({aP}-1) }$ and $\large \frac{dv}{dx}= \simplify{{eP}*e^({eP} x)}$

We now use the formula:

$\large \frac{dy}{dx}=u \frac{dv}{dx} + v \frac{du}{dx}$

Make the appropriate substitutions into the formula:

$\large \frac{dy}{dx}= \var{aCF}x^{\var{aP}} \times \simplify{{eP}*e^({eP} x)} + e^{\var{eP}x} \times \simplify{{aP}*{aCF}*x^({aP}-1) }$

Finally, we need to use our basic algebra to simplify this as much as possible. Multiply out any brackets where it would simplify and collect like terms:

$\large \frac{dy}{dx}= \simplify{{aCF}x^{{aP}} * {eP}*e^({eP} x)} + \simplify{{aP}*{aCF}*x^{{aP}-1} } e^{\var{eP}x}$

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
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exam:

Product Rule (Instructional) by Michael Proudman in .Differential Calculus.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \times v$ then:

$\frac{dy}{dx}=u \frac{dv}{dx} + v \frac{du}{dx}$

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Differentiate $y=\var{aCF}x^{\var{aP}} e^{\var{eP}x}$

Loading...

Error

Product Rule 05

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

Michael Proudman 4 years, 10 months ago

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If y=u×vy=u \times v then:

dydx=udvdx+vdudx \frac{dy}{dx}=u \frac{dv}{dx} + v \frac{du}{dx}

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Differentiate y={aCF}x{aP}e{eP}x y=\var{aCF}x^{\var{aP}} e^{\var{eP}x}

If $y=u \times v$ then:

$\frac{dy}{dx}=u \frac{dv}{dx} + v \frac{du}{dx}$

Generated value: `integer`

Differentiate $y=\var{aCF}x^{\var{aP}} e^{\var{eP}x}$