Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\sin (\sqrt[3]{x}) + \sqrt[3]{\sin (x)}$ with respect to $x$ is $\frac{d}{d x} [\sin (\sqrt[3]{x})] + \frac{d}{d x} [\sqrt[3]{\sin (x)}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\sin (\sqrt[3]{x})]$ .

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{3}}$ .

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

Step 2.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [x^{\frac{1}{3}}] + \frac{d}{d x} [\sqrt[3]{\sin (x)}]$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{3}}$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

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

Step 2.7.2

Subtract $3$ from $1$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 3

Evaluate $\frac{d}{d x} [\sqrt[3]{\sin (x)}]$ .

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Step 3.1

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

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

Step 3.2

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 3.2.1

To apply the Chain Rule, set $u_{2}$ as $\sin (x)$ .

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{3}$ .

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

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $3$ .

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

Step 3.7.2

Subtract $3$ from $1$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Separate fractions.

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Reorder factors in $\frac{\cos (x^{\frac{1}{3}})}{3 x^{\frac{2}{3}}} + \frac{\cos (x) {\csc (x)}^{\frac{2}{3}}}{3}$ .

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