Calculus Examples

$\sqrt[3]{\sin (x)}$

Step 1

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

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

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 $g (x) = \sin (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Combine $\frac{1}{3}$ and ${\sin (x)}^{- \frac{2}{3}}$ .

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

Step 7.3

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

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

Step 8

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{1}{3 {\sin (x)}^{\frac{2}{3}}} \cos (x)$

Step 9

Combine $\frac{1}{3 {\sin (x)}^{\frac{2}{3}}}$ and $\cos (x)$ .

$\frac{\cos (x)}{3 {\sin (x)}^{\frac{2}{3}}}$

Step 10

Simplify.

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Step 10.1

Separate fractions.

$\frac{\cos (x)}{3} \cdot \frac{1}{{\sin (x)}^{\frac{2}{3}}}$

Step 10.2

Convert from $\frac{1}{{\sin (x)}^{\frac{2}{3}}}$ to ${\csc (x)}^{\frac{2}{3}}$ .

$\frac{\cos (x)}{3} {\csc (x)}^{\frac{2}{3}}$

Step 10.3

Combine $\frac{\cos (x)}{3}$ and ${\csc (x)}^{\frac{2}{3}}$ .

$\frac{\cos (x) {\csc (x)}^{\frac{2}{3}}}{3}$

Step 10.4

Reorder factors in $\frac{\cos (x) {\csc (x)}^{\frac{2}{3}}}{3}$ .

$\frac{{\csc (x)}^{\frac{2}{3}} \cos (x)}{3}$