Chain Rule 06

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Michael Proudman

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Needs to be tested

4 years, 10 months ago

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Michael Proudman 4 years, 10 months ago

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Name	Status	Author	Last Modified
Chain Rule 06	Needs to be tested	Michael Proudman	07/12/2021 08:32
Chain Rule 06 (non-scaffolded)	Needs to be tested	Michael Proudman	07/12/2021 08:32

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Question statement

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function "nested" inside another.

If $y=f(g(x))$ to find $\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

$\large \frac{dy}{dx}=\frac{du}{dx} \times \frac{dy}{du}$

Give any introductory information the student needs.

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Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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3. Gap Gap 2. Mathematical expression
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4. Gap Gap 3. Mathematical expression
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5. Gap Gap 4. Mathematical expression
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6. Gap Gap 5. Mathematical expression
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7. Gap Gap 6. Mathematical expression
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8. Gap Gap 7. Mathematical expression
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Ask the student a question, and give any hints about how they should answer this part.

Differentiate $y=\ln({\var{CF}x+\sin{(x)})}$

First identify the "innermost" function, and substitute $u$ :

Let $u=$ Gap 0. Then $y=$ Gap 1.

Then:

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

Now using:

$\large \frac{dy}{dx}=\frac{du}{dx} \times \frac{dy}{du}$

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

Which simplifies to:

$\large \frac{dy}{dx}=$ Gap 6.

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\large \frac{dy}{dx}=$ Gap 7.

Advice

We are asked to differentiate:

$y=\ln({\simplify{{CF}x}+\sin{(x)})}$

Recognising that this is a "function of another function", we need to identify the "innermost" of the two functions that are involved and substitute $u$

Let $u=\simplify{{CF}x+sin(x)}$ then $y=\ln(u)$

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule and your Table of Derivatives.:

Applying this method gives us:

$\large \frac{du}{dx}=\var{CF}+cos(x)$ and $\large \frac{dy}{du}= \frac{1}{u}$

We now use the Chain Rule formula:

$\large \large \frac{dy}{dx}=\frac{du}{dx} \times \frac{dy}{du}$

Make the appropriate substitutions into the formula:

$\large \frac{dy}{dx}= (\var{CF}+cos(x)) \times \frac{1}{u}$

Which simplifies to:

$\large \frac{dy}{dx}=\frac{\var{CF}+cos(x)}{u}$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\large \frac{dy}{dx}=\frac{\var{CF}+cos(x)}{\simplify{{CF}x+sin(x)}}$

Give a worked solution to the whole question.

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This question is used in the following exam:

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

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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 CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function "nested" inside another.

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Differentiate $y=\ln({\var{CF}x+\sin{(x)})}$

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Error

Chain Rule 06

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 CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function "nested" inside another.

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Differentiate y=ln({CF}x+sin(x)) y=\ln({\var{CF}x+\sin{(x)})}

Generated value: `integer`

Differentiate $y=\ln({\var{CF}x+\sin{(x)})}$