Quotient 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

3 years, 1 month ago

History

○

Michael Proudman 3 years, 1 month ago

Gave some feedback: Needs to be tested

○

Michael Proudman 4 years, 11 months ago

Published this.

○

Michael Proudman 4 years, 11 months ago

Created this as a copy of Quotient Rule 04.

Name	Status	Author	Last Modified
Quotient Rule 01	Ready to use	Michael Proudman	07/03/2022 08:07
Quotient Rule 02	Needs to be tested	Michael Proudman	15/03/2022 16:35
Quotient Rule 03	Needs to be tested	Michael Proudman	15/03/2022 16:43
Quotient Rule 04	Needs to be tested	Michael Proudman	15/03/2022 16:48
Quotient Rule 05	Needs to be tested	Michael Proudman	15/03/2022 16:48
Quotient Rule 06	Needs to be tested	Michael Proudman	15/03/2022 16:50
Quotient Rule 01 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 02 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 03 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 04 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 05 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32
Quotient Rule 06 (non-scaffolded)	draft	Michael Proudman	07/12/2021 08:32

There are 5 other versions that do you not have access to.

Question statement

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

If $\large y=\frac{u}{v}$ then:

$\frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

			aC1	integer	3
			aP	integer	4

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

A random number between and (inclusive) with step size

Description

Part a) Constant term for denominator

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:

This variable doesn't seem to be used anywhere.

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
10. Gap Gap 9. Mathematical expression
  Add

Gap-fill

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

Differentiate $\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$

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

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

Now differentiate each one:

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

Then using:

$\Large \frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Substitute each component into the formula in the correct place:

$\Large \frac{dy}{dt}=$

Gap 4.

$\times$ Gap 5.

$\large -$ Gap 6.

$\times$ Gap 7.Gap 8.

Finally tidy this up to give your final answer:

$\Large \frac{dy}{dx}=$ Gap 9.

Advice

We are asked to differentiate:

$\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

$u$ is the numerator, the function "on top", $v$ is the denominator, the function "on the bottom".

$u=e^{\var{aP}t}$ $v=t+\var{aC1}$

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

if $\frac{d}{dx} [a^x] = a^x \ln{(a)}$

If you don't follow this, use your Table of Derivatives.

Applying this gives us:

$\large \frac{du}{dx}=\simplify{{aP}e^({aP}t)}$ and $\frac{dv}{dx}=1$

Make the appropriatesubstitutions into the formula:

$\large \frac{dy}{dx}= \frac{(t+\var{aC1}) \times (\simplify{{aP}e^({aP}t)}) - e^{\var{aP}t} }{(t+\var{aC1})^2}$

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

$\large \frac{dy}{dx}= \frac{\simplify{{aP}e^({aP}t)}(t+\var{aC1}) - (e^{\var{aP}t}) }{(t+\var{aC1})^2}$

Give a worked solution to the whole question.

URL
resources/question-resources/

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
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exam:

Quotient 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 3 years, 1 month ago

Michael Proudman 4 years, 11 months ago

Michael Proudman 4 years, 11 months ago

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

If $\large y=\frac{u}{v}$ then:

$\frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Differentiate $\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$

Loading...

Error

Quotient Rule 05

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Michael Proudman 3 years, 1 month ago

Michael Proudman 4 years, 11 months ago

Michael Proudman 4 years, 11 months ago

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

If y=uv\large y=\frac{u}{v} then:

dydx=vdudx−udvdxv2 \frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Differentiate y=e{aP}tt+{aC1} \large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}

If $\large y=\frac{u}{v}$ then:

$\frac{dy}{dx}= \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Generated value: `integer`

Differentiate $\large y=\frac{e^{\var{aP}t}}{t+\var{aC1}}$