Power Rule 4

Question 1 of 4

Find the derivative

f (x) = \frac{6}{x \sqrt{x}}

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

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

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

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

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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)

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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, 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 $0$
	$=$	$4x+8$

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

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Quiz summary

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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

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