Power Rule 2

Question 1 of 5

Find the derivative

f (x) = 2 x - 5

$f(x)=2x-5$

(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:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{2x}^{\color{#e65021}{1}}-5$

$x$	$=$	$2x$
$n$	$=$	$1$

Second term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$2x-\color{#9a00c7}{5}$

$x$

$=$

$5$

Substitute the values into the power rule

First term:

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

Second term:

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

$f'(x)=2$

Question 2 of 5

Find the derivative

$f(x)=1/2 x^6$

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

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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

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

$f(x)$	$=$	$\color{#9a00c7}{\frac{1}{2}x}^{\color{#e65021}{6}}$

$x$	$=$	$1/2 x$

$n$	$=$	$6$

Substitute the values into the power rule

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

	$=$	$\color{#e65021}{6}\cdot\color{#9a00c7}{\frac{1}{2}x}^{\color{#e65021}{6}-1}$	Substitute known values

	$=$	$3x^5$

$f'(x)=3x^5$

Question 3 of 5

Find the derivative

$f(x)=1/(4x^5)$

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

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Power Rule

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

First, reciprocate

$x^5$ to remove the exponent from the denominator

$f(x)$	$=$	$1/(4x^5)$

	$=$	$1/4 x^(-5)$	Reciprocate $x^5$

Next, identify the values of the function

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

$f(x)$	$=$	$\color{#9a00c7}{\frac{1}{4}x}^{\color{#e65021}{-5}}$

$x$	$=$	$1/4 x$

$n$	$=$	$-5$

Substitute the values into the power rule

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

	$=$	$\color{#e65021}{-5}\cdot\color{#9a00c7}{\frac{1}{4}x}^{\color{#e65021}{-5}-1}$	Substitute known values

	$=$	$-5/4 x^(-6)$

	$=$	$-5/(4x^6)$	Reciprocate $x^(-6)$

$f'(x)=-5/(4x^6)$

Question 4 of 5

Find the derivative

$f(x)=2/(3x^3)$

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

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Power Rule

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

First, reciprocate

$x^3$ to remove the exponent from the denominator

$f(x)$	$=$	$2/(3x^3)$

	$=$	$2/3 x^(-3)$	Reciprocate $x^5$

Next, identify the values of the function

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

$f(x)$	$=$	$\color{#9a00c7}{\frac{2}{3}x}^{\color{#e65021}{-3}}$

$x$	$=$	$2/3 x$

$n$	$=$	$-3$

Substitute the values into the power rule

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

	$=$	$\color{#e65021}{-3}\cdot\color{#9a00c7}{\frac{2}{3}x}^{\color{#e65021}{-3}-1}$	Substitute known values

	$=$	$-2 x^(-4)$

	$=$	$-2/(x^4)$	Reciprocate $x^(-4)$

$f'(x)=-2/(x^4)$

Question 5 of 5

Find the derivative

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

1. $f'(x)=-4/(x^5)$
2. $f'(x)=-(12)/(x^4)$
3. $f'(x)=(12)/(x^5)$
4. $f'(x)=-(12)/(x^5)$

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Power Rule

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

First, reciprocate

$x^4$ to remove the exponent from the denominator

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

	$=$	$3x^(-4)$	Reciprocate $x^(3/2)$

Next, identify the values of the function

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

$x$	$=$	$3x$
$n$	$=$	$-4$

Substitute the values into the power rule

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{-4}\cdot\color{#9a00c7}{3x}^{\color{#e65021}{-4}-1}$	Substitute known values
	$=$	$-12x^(-5)$

	$=$	$-(12)/(x^5)$	Reciprocate $x^(-5)$

$f'(x)=-(12)/(x^5)$

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Quiz summary

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1. Question

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Nice Job!

Power Rule

Remember

2. Question

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Well Done!

Power Rule

3. Question

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Correct!

Power Rule

4. Question

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Fantastic!

Power Rule

5. Question

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Correct!

Power Rule

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