Calculus Examples

$\sqrt[3]{\cos^{2} (3 x)}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\cos^{2} (3 x)}$ as ${\cos (3 x)}^{\frac{2}{3}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = \cos (3 x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (3 x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\cos (3 x)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $2$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

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

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Step 8.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

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

Step 8.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

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

Step 8.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 9

Differentiate using the Constant Multiple Rule.

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Step 9.1

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

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

Step 9.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 9.3

Simplify terms.

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Step 9.3.1

Multiply $3$ by $- 1$ .

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

Step 9.3.2

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

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

Step 9.3.3

Multiply $2$ by $- 3$ .

$\frac{- 6 \sin (3 x)}{3 {\cos (3 x)}^{\frac{1}{3}}} \frac{d}{d x} [x]$

Step 9.3.4

Factor $3$ out of $- 6 \sin (3 x)$ .

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

Step 10

Cancel the common factors.

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Step 10.1

Factor $3$ out of $3 {\cos (3 x)}^{\frac{1}{3}}$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

Multiply $- 1$ by $1$ .

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