Chain rule. (Video).

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Bill Foster 55 years, 2 months ago

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Question statement

Differentiate the following function $f(x)$ using the chain rule.

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a	integer	3
s2	integer	1
c	integer	9
b	integer	1
s1	integer	1

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$\simplify[std]{f(x) = cos(e^({a}x) +{b}x^2+{c})}$

$\displaystyle \frac{df}{dx}=\;$ Unnamed gap

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Advice

$\simplify[std]{f(x) = cos(e^({a}x) +{b}x^2+{c})}$
The chain rule says that if $f(x)=g(h(x))$ then

$\simplify[std]{f'(x) = h'(x)g'(h(x))}$
One way to find

$f'(x)$ is to let

$u=h(x)$ then we have

$f(u)=g(u)$ as a function of

$u$ .
Then we use the chain rule in the form:

$\frac{df}{dx} = \frac{du}{dx}\frac{df(u)}{du}$
Once you have worked this out, you replace

$u$ by

$h(x)$ and your answer is now in terms of

$x$ .

For this example, we let $u=\simplify[std]{e^({a}x) +{b}x^2+{c}}$ and we have $f(u)=\simplify[std]{cos(u)}$ .
This gives

$\begin{eqnarray*}\frac{du}{dx} &=& \simplify[std]{{a}e^({a}x) +{2*b}x}\\ \frac{df(u)}{du} &=& \simplify[std]{-sin(u)} \end{eqnarray*}$

Hence on substituting into the chain rule above we get:

$\begin{eqnarray*}\frac{df}{dx} &=& \simplify[std]{({a}e^({a}x) +{2*b}x) * (-sin(u))}\\ &=& \simplify[std]{-({a}e^({a}x) +{2*b}x)*sin(e^({a}x) +{b}x^2+{c})} \end{eqnarray*}$
on replacing

$u$ by

$\simplify[std]{e^({a}x) +{b}x^2+{c}}$ .

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This question is used in the following exams:

mathcentre: Chain rule practice by Bill Foster in Bill's workspace.
First Exam by Sarah Jowett in Sarah's workspace.
maths methods week 9 by cormac murphy in cormac's workspace.
Bradford: Diagnostic test - Basic calculus by Bill Foster in Bill's workspace.
Lois's copy of Bradford: Diagnostic test - Basic calculus by Lois Rollings in Lois's workspace.
Chain rule by Peter Chapman in Peter's workspace.
Maths Support: Chain rule practice by Christian Lawson-Perfect in Maths Support Wiki.
Tore's copy of mathcentre: Chain rule practice by Tore Gaupseth in Tore's workspace.
CFM Chain rule by Jinhua Mathias in Deactivated user's workspace.
Basic Calculus by Bill Foster in Bill's workspace.
Bradford: Basic Calculus by Bill Foster in Bill's workspace.
Stanislav's copy of Bradford: Basic Calculus by Stanislav Senveter in Stanislav's workspace.
mathcentre: Diagnostic test - Basic calculus by Bill Foster in Bill's workspace.
Chain Rule by Bill Foster in Bill's workspace.
Differentiation by Bill Foster in Bill's workspace.
Simon's copy of Maths Support: Chain rule practice by Simon Thomas in Calculus.

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Chain rule. (Video).

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`