Chain Rule 1

Question 1 of 4

Find the derivative using the power rule

$y=(5x+2)^3$

1. $f'(y)=15(5x+2)^3$
2. $f'(y)=15(5x+2)^2$
3. $f'(y)=15(5x-2)^2$
4. $f'(y)=3(5x+2)^2$

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Power Rule for Function within a Function

$\frac{dy}{dx}=\color{#e65021}{\frac{dy}{du}}\cdot\color{#004ec4}{\frac{du}{dx}}$

$f'(y)=(dy)/dx$

$y’=$ $(dy)/(du)$

$u’=$ $(du)/dx$

First, find the derivative of $u$ and $y$ with respect to

$u$ .

Derivative of

$y$ with respect to

$u$ :

$y$	$=$	$u^(3)$
$y’$	$=$	$3u^2$

Derivative of

$u$ :

$u$	$=$	$5x+2$
$u’$	$=$	$5$

Substitute the components into the product rule

$y’$	$=$	$(dy)/(du)$	$=$	$3u^2$

$u’$	$=$	$(du)/(dx)$	$=$	$5$

$\frac{dy}{dx}$	$=$	$\color{#e65021}{\frac{dy}{du}}\cdot\color{#004ec4}{\frac{du}{dx}}$

$f'(y)$	$=$	$\color{#e65021}{3u^2}\cdot\color{#004ec4}{5}$	Substitute known values
	$=$	$15u^2$
	$=$	$15(5x+2)^2$	Substitute $u=5x+2$

$f'(y)=15(5x+2)^2$

Question 2 of 4

Find the derivative using the chain rule

$f(x)=7(3x+4)^5$

1. $y'=35(3x+4)^4$
2. $y'=105(3x-4)^5$
3. $y'=105(3x+4)^4$
4. $y'=105(3x+4)^5$

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Chain Rule

$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$

First, identify the values of the function

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$7(\color{#9a00c7}{3x+4})^{\color{#e65021}{5}}$

$x$	$=$	$3x+4$
$n$	$=$	$5$

Finally, substitute the values into the chain rule

$y’$	$=$	$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$
	$=$	$\color{#e65021}{5}\cdot7(\color{#004ec4}{3x+4})^{\color{#e65021}{(5)}-1}\cdot\color{#00880A}{f'(3x+4)}$	Substitute known values
	$=$	$35(3x+4)^{4}\cdot\color{#00880A}{3}$	Differentiate $5x+2$
	$=$	$105(3x+4)^{4}$	Evaluate

$y’=105(3x+4)^4$

Question 3 of 4

Find the derivative using the chain rule

$f(x)=(4x-5)^7$

1. $y'=7(4x-5)^6$
2. $y'=28(4x-5)^6$
3. $y'=28(4x+5)^6$
4. $y'=28(4x-5)^7$

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Chain Rule

$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$

First, identify the values of the function

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$(\color{#9a00c7}{4x-5})^{\color{#e65021}{7}}$

$x$	$=$	$4x-5$
$n$	$=$	$7$

Finally, substitute the values into the chain rule

$y’$	$=$	$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$
	$=$	$\color{#e65021}{7}\cdot(\color{#004ec4}{4x-5})^{\color{#e65021}{(7)}-1}\cdot\color{#00880A}{f'(4x-5)}$	Substitute known values
	$=$	$7(4x-5)^{6}\cdot\color{#00880A}{4}$	Differentiate $4x-5$
	$=$	$28(4x-5)^{6}$	Evaluate

$y’=28(4x-5)^6$

Question 4 of 4

Find the derivative

$f(x)=5/(3x-6)$

1. $y'=-15/((3x-6)^2)$
2. $y'=-3/((3x-6)^2)$
3. $y'=15/((3x-6)^2)$
4. $y'=15/(3x-6)$

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Chain Rule

$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$

First, remove the fraction by reciprocating the denominator

$5/(3x-6)$

$=$

$5(3x-6)^(-1)$

Next, identify the values of the function

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{3x-6}^{\color{#e65021}{-1}}$

$x$	$=$	$3x-6$
$n$	$=$	$-1$

Finally, substitute the values into the chain rule

$y’$	$=$	$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$
	$=$	$\color{#e65021}{-1}\cdot5(\color{#004ec4}{3x-6})^{\color{#e65021}{(-1)}-1}\cdot\color{#00880A}{f'(3x-6)}$	Substitute known values
	$=$	$-5(3x-6)^{-2}\cdot\color{#00880A}{3}$	Differentiate $3x-6$
	$=$	$-15(3x-6)^{-2}$	Evaluate

	$=$	$-15/((3x-6)^2)$	Reciprocate $(3x-6)^(-2)$

$y’=-15/((3x-6)^2)$

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Quiz summary

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1. Question

Hint

Well Done!

Power Rule for Function within a Function

2. Question

Hint

Chain Rule

3. Question

Hint

Fantastic!

Chain Rule

4. Question

Hint

Keep Going!

Chain Rule

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