Power Rule 4

Question 1 of 4

Find the derivative

$f(x)=6/(xsqrtx)$

1. $f'(x)=-9x^(5/2)$
2. $f'(x)=9x^(-5/2)$
3. $f'(x)=-9x^(-5/2)$
4. $f'(x)=9x^(-5/2)$

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Power Rule

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

First, convert the surd into an exponent.

$f(x)$	$=$	$6/(xsqrtx)$

	$=$	$6/(x^1 times x^(1/2))$	Convert the surd into an exponent

	$=$	$6/(x^(3/2))$

Next, reciprocate

$x^(3/2)$ to remove the exponent from the denominator

$f(x)$	$=$	$6/(x^(3/2))$

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

Next, identify the values of the function

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{6x}^{\color{#e65021}{-\frac{3}{2}}}$

$x$	$=$	$6x$
$n$	$=$	$-3/2$

Substitute the values into the power rule

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

	$=$	$\color{#e65021}{-\frac{3}{2}}\cdot\color{#9a00c7}{6x}^{\color{#e65021}{-\frac{3}{2}}-1}$	Substitute known values

	$=$	$-9x^(-5/2)$

$f'(x)=-9x^(-5/2)$

Question 2 of 4

Find the derivative

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

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

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Power Rule

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

First, convert the surd into an exponent.

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

	$=$	$(x^(1/2))/(x^2)$	Convert the surd into an exponent

Next, reciprocate

$x^2$ to remove the exponent from the denominator

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

	$=$	$x^(1/2) times x^(-2)$	Reciprocate $x^2$

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

Next, identify the values of the function

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

$f(x)$	$=$	$\color{#9a00c7}{6x}^{\color{#e65021}{-\frac{3}{2}}}$

$x$	$=$	$x$

$n$	$=$	$-3/2$

Substitute the values into the power rule

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

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

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

$f'(x)=-3/2 x^(-5/2)$

Question 3 of 4

Find the derivative

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

1. $f'(x)=5x^6-4x^7$
2. $f'(x)=6x^5+35x^4$
3. $f'(x)=6x^6-35x^5$
4. $f'(x)=6x^5-35x^4$

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Power Rule

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

First, distribute

$x^5$ to the value inside the parenthesis.

$f(x)$	$=$	$x^5(x-7)$
	$=$	$(xtimesx^5)-(7timesx^5)$	Distribute
	$=$	$x^6-7x^5$

Next, identify the values of the function

First term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{6}}-7x^5$

$x$	$=$	$x$
$n$	$=$	$6$

Second term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$x^6-\color{#9a00c7}{7x}^{\color{#e65021}{5}}$

$x$	$=$	$7x$
$n$	$=$	$5$

Substitute the values into the power rule

First term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{6}\cdot\color{#9a00c7}{x}^{\color{#e65021}{6}-1}-7x^5$	Substitute known values
	$=$	$6x^5-7x^5$

Second term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$6x^5-\color{#e65021}{5}\cdot\color{#9a00c7}{7x}^{\color{#e65021}{5}-1}$	Substitute known values
	$=$	$6x^5-35x^4$

$f'(x)=6x^5-35x^4$

Question 4 of 4

Find the derivative

$f(x)=(x+3)(2x+2)$

$f'(x)=$ (4x+8)

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

Power Rule

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

Remember

Differentiating a constant makes it

$0$ .

First, expand the equation.

$f(x)$	$=$	$(x+3)(2x+2)$
	$=$	$2x^2+2x+6x+6$	Expand
	$=$	$2x^2+8x+6$	Combine like terms

Next, identify the values of the function

First term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$\color{#9a00c7}{2x}^{\color{#e65021}{2}}+8x+6$

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

Second term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$2x^2+\color{#9a00c7}{8x}+6$

$x$	$=$	$8x$
$n$	$=$	$1$

Third term:

$f(x)$	$=$	$\color{#9a00c7}{x}^{\color{#e65021}{n}}$
$f(x)$	$=$	$2x^2+8x+\color{#9a00c7}{6}$

$x$

$=$

$6$

Substitute the values into the power rule

First term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$\color{#e65021}{2}\cdot\color{#9a00c7}{2x}^{\color{#e65021}{2}-1}+8x+6$	Substitute known values
	$=$	$4x+8x+6$

Second term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$4x+\color{#e65021}{1}\cdot\color{#9a00c7}{8}^{\color{#e65021}{1}-1}+6$	Substitute known values
	$=$	$4x+8+6$

Third term:

$f'(x)$	$=$	$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$
	$=$	$4x+8+\color{#9a00c7}{0}$	Differentiating a constant makes it $0$
	$=$	$4x+8$

$f'(x)=4x+8$

Try VividMath Premium to unlock full access

Quiz summary

Information

1. Question

Hint

Keep Going!

Power Rule

2. Question

Hint

Correct!

Power Rule

3. Question

Hint

Great Work!

Power Rule

4. Question

Hint

Exceptional!

Power Rule

Remember

Quizzes