Calculus Examples

$\sqrt[3]{\frac{8}{x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{8}{x}}$ as ${(\frac{8}{x})}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(\frac{8}{x})}^{\frac{1}{3}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = \frac{8}{x}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $\frac{8}{x}$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [\frac{8}{x}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{3}$ .

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [\frac{8}{x}]$

Step 2.3

Replace all occurrences of $u$ with $\frac{8}{x}$ .

$\frac{1}{3} {(\frac{8}{x})}^{\frac{1}{3} - 1} \frac{d}{d x} [\frac{8}{x}]$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} {(\frac{8}{x})}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [\frac{8}{x}]$

Step 4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{1}{3} {(\frac{8}{x})}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [\frac{8}{x}]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{3} {(\frac{8}{x})}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [\frac{8}{x}]$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(\frac{8}{x})}^{\frac{1 - 3}{3}} \frac{d}{d x} [\frac{8}{x}]$

Step 6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(\frac{8}{x})}^{\frac{- 2}{3}} \frac{d}{d x} [\frac{8}{x}]$

Step 7

Move the negative in front of the fraction.

$\frac{1}{3} {(\frac{8}{x})}^{- \frac{2}{3}} \frac{d}{d x} [\frac{8}{x}]$

Step 8

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{3} {(\frac{8}{x})}^{- \frac{2}{3}} (8 \frac{d}{d x} [\frac{1}{x}])$

Step 9

Combine fractions.

Tap for more steps...

Step 9.1

Combine $8$ and $\frac{1}{3}$ .

$\frac{8}{3} {(\frac{8}{x})}^{- \frac{2}{3}} \frac{d}{d x} [\frac{1}{x}]$

Step 9.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{8}{3} {(\frac{8}{x})}^{- \frac{2}{3}} \frac{d}{d x} [x^{- 1}]$

Step 10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 1$ .

$\frac{8}{3} {(\frac{8}{x})}^{- \frac{2}{3}} (- x^{- 2})$

Step 11

Combine fractions.

Tap for more steps...

Step 11.1

Combine $\frac{8}{3}$ and $x^{- 2}$ .

${(\frac{8}{x})}^{- \frac{2}{3}} (- \frac{8 x^{- 2}}{3})$

Step 11.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{8}{x})}^{- \frac{2}{3}} (- \frac{8}{3 x^{2}})$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{x}{8})}^{\frac{2}{3}} (- \frac{8}{3 x^{2}})$

Step 12.2

Apply the product rule to $\frac{x}{8}$ .

$\frac{x^{\frac{2}{3}}}{8^{\frac{2}{3}}} (- \frac{8}{3 x^{2}})$

Step 12.3

Combine terms.

Tap for more steps...

Step 12.3.1

Rewrite $8$ as $2^{3}$ .

$\frac{x^{\frac{2}{3}}}{{(2^{3})}^{\frac{2}{3}}} (- \frac{8}{3 x^{2}})$

Step 12.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{\frac{2}{3}}}{2^{3 (\frac{2}{3})}} (- \frac{8}{3 x^{2}})$

Step 12.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 12.3.3.1

Cancel the common factor.

$\frac{x^{\frac{2}{3}}}{2^{3 (\frac{2}{3})}} (- \frac{8}{3 x^{2}})$

Step 12.3.3.2

Rewrite the expression.

$\frac{x^{\frac{2}{3}}}{2^{2}} (- \frac{8}{3 x^{2}})$

Step 12.3.4

Raise $2$ to the power of $2$ .

$\frac{x^{\frac{2}{3}}}{4} (- \frac{8}{3 x^{2}})$

Step 12.3.5

Multiply $\frac{x^{\frac{2}{3}}}{4}$ by $\frac{8}{3 x^{2}}$ .

$- \frac{x^{\frac{2}{3}} \cdot 8}{4 (3 x^{2})}$

Step 12.3.6

Multiply $3$ by $4$ .

$- \frac{x^{\frac{2}{3}} \cdot 8}{12 x^{2}}$

Step 12.3.7

Move $8$ to the left of $x^{\frac{2}{3}}$ .

$- \frac{8 \cdot x^{\frac{2}{3}}}{12 x^{2}}$

Step 12.3.8

Move $x^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{8}{12 x^{2} x^{- \frac{2}{3}}}$

Step 12.3.9

Multiply $x^{2}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

Tap for more steps...

Step 12.3.9.1

Move $x^{- \frac{2}{3}}$ .

$- \frac{8}{12 (x^{- \frac{2}{3}} x^{2})}$

Step 12.3.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{8}{12 x^{- \frac{2}{3} + 2}}$

Step 12.3.9.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{8}{12 x^{- \frac{2}{3} + 2 \cdot \frac{3}{3}}}$

Step 12.3.9.4

Combine $2$ and $\frac{3}{3}$ .

$- \frac{8}{12 x^{- \frac{2}{3} + \frac{2 \cdot 3}{3}}}$

Step 12.3.9.5

Combine the numerators over the common denominator.

$- \frac{8}{12 x^{\frac{- 2 + 2 \cdot 3}{3}}}$

Step 12.3.9.6

Simplify the numerator.

Tap for more steps...

Step 12.3.9.6.1

Multiply $2$ by $3$ .

$- \frac{8}{12 x^{\frac{- 2 + 6}{3}}}$

Step 12.3.9.6.2

Add $- 2$ and $6$ .

$- \frac{8}{12 x^{\frac{4}{3}}}$

Step 12.3.10

Factor $4$ out of $8$ .

$- \frac{4 (2)}{12 x^{\frac{4}{3}}}$

Step 12.3.11

Cancel the common factors.

Tap for more steps...

Step 12.3.11.1

Factor $4$ out of $12 x^{\frac{4}{3}}$ .

$- \frac{4 (2)}{4 (3 x^{\frac{4}{3}})}$

Step 12.3.11.2

Cancel the common factor.

$- \frac{4 \cdot 2}{4 (3 x^{\frac{4}{3}})}$

Step 12.3.11.3

Rewrite the expression.

$- \frac{2}{3 x^{\frac{4}{3}}}$