Power Rule 1

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AP Calculus BC

Derivatives

Power Rule

Power Rule 1

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Question 1 of 5

Find the derivative

$f(x)=x^8$

1. $f'(x)=8x^8$
2. $f'(x)=8x^7$
3. $f'(x)=8x$
4. $f'(x)=7x^8$

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Power Rule

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

First, identify the values of the function

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{8}}$

$x$	$=$	$x$
$n$	$=$	$8$

Substitute the values into the power rule

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{8}\cdot\color{#9a00c7}{x}^{\color{#e65021}{8}-1}$	Substitute known values
	$=$	$8x^7$

$f'(x)=8x^7$

Question 2 of 5

Find the derivative

$f(x)=3x$

1. $f'(x)=3x^3$
2. $f'(x)=3$
3. $f'(x)=3x$
4. $f'(x)=x^3$

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Power Rule

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

First, identify the values of the function

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

$x$	$=$	$3x$
$n$	$=$	$1$

Substitute the values into the power rule

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{1}\cdot\color{#9a00c7}{3x}^{\color{#e65021}{1}-1}$	Substitute known values
	$=$	$3x^0$
	$=$	$3xx1$	Anything raised to $0$ is $1$
	$=$	$3$

$f'(x)=3$

Question 3 of 5

Find the derivative

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

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

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Power Rule

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

First, identify the values of the function

First term:

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

$x$	$=$	$x$
$n$	$=$	$3$

Second term:

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

$x$	$=$	$x$
$n$	$=$	$4$

Substitute the values into the power rule

First term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{3}\cdot\color{#9a00c7}{x}^{\color{#e65021}{3}-1}+x^4$	Substitute known values
	$=$	$3x^2+x^4$

Second term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$3x^2+\color{#e65021}{4}\cdot\color{#9a00c7}{x}^{\color{#e65021}{4}-1}$	Substitute known values
	$=$	$3x^2+4x^3$

$f'(x)=3x^2+4x^3$

Question 4 of 5

4. Question
Find the derivative

$f(x)=8$
- $f'(x)=$ (0)
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Power Rule

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

Remember

Differentiating a constant makes it $0$ .

First, identify the values of the function

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

$f(x)$ $=$ $\color{#9a00c7}{8}$

$x$ $=$ $8$

The value of the function is a constant.

Since differentiating a constant makes it $0$ , the derivative of this function is $0$ .

$f'(x)=0$

Question 5 of 5

Find the derivative

$f(x)=x^5+x^3+6$

1. $f'(x)=5x^5+3x^3+6$
2. $f'(x)=4x^5+2x^3$
3. $f'(x)=5x^4+3x^2+6$
4. $f'(x)=5x^4+3x^2$

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Power Rule

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

Remember

Differentiating a constant makes it

$0$ .

First, identify the values of the function

First term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{5}}+x^3+6$

$x$	$=$	$x$
$n$	$=$	$5$

Second term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$x^5+\color{#9a00c7}{x}^{\color{#e65021}{3}}+6$

$x$	$=$	$x$
$n$	$=$	$3$

Third term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$x^5+x^3+\color{#9a00c7}{6}$

$x$

$=$

$6$

Substitute the values into the power rule

First term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{5}\cdot\color{#9a00c7}{x}^{\color{#e65021}{5}-1}+x^3+6$	Substitute known values
	$=$	$5x^4+x^3+6$

Second term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$5x^4+\color{#e65021}{3}\cdot\color{#9a00c7}{x}^{\color{#e65021}{3}-1}+6$	Substitute known values
	$=$	$5x^4+3x^2+6$

Third term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$5x^4+3x^2+\color{#9a00c7}{0}$	Differentiating a constant makes it $0$
	$=$	$5x^4+3x^2$

$f'(x)=5x^4+3x^2$

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Quiz summary

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1. Question

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Great Work!

Power Rule

2. Question

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Correct!

Power Rule

3. Question

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Keep Going!

Power Rule

4. Question

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Fantastic!

Power Rule

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5. Question

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Excellent!

Power Rule

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