Calculus Examples

$f (x) = \ln (\cos (\sqrt[3]{x + 2}))$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 2}$ as ${(x + 2)}^{\frac{1}{3}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \cos ({(x + 2)}^{\frac{1}{3}})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cos ({(x + 2)}^{\frac{1}{3}})$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\cos ({(x + 2)}^{\frac{1}{3}})$ .

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

Step 3

Convert from $\frac{1}{\cos ({(x + 2)}^{\frac{1}{3}})}$ to $\sec ({(x + 2)}^{\frac{1}{3}})$ .

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

Step 4

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 $g (x) = {(x + 2)}^{\frac{1}{3}}$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as ${(x + 2)}^{\frac{1}{3}}$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u_{2}$ with ${(x + 2)}^{\frac{1}{3}}$ .

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

Step 5

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 $g (x) = x + 2$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $3$ .

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

Step 9.2

Subtract $3$ from $1$ .

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

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

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

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

Step 11

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

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

Step 12

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

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

Step 13

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- \frac{\sec ({(x + 2)}^{\frac{1}{3}}) \sin ({(x + 2)}^{\frac{1}{3}})}{3 {(x + 2)}^{\frac{2}{3}}} (1 + 0)$

Step 14

Simplify the expression.

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Step 14.1

Add $1$ and $0$ .

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

Step 14.2

Multiply $- 1$ by $1$ .

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

Step 15

Simplify.

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Step 15.1

Rewrite $\sec ({(x + 2)}^{\frac{1}{3}})$ in terms of sines and cosines.

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

Step 15.2

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

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

Step 15.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 15.4

Combine.

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

Step 15.5

Multiply $\sin ({(x + 2)}^{\frac{1}{3}})$ by $1$ .

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

Step 15.6

Move $3$ to the left of $\cos ({(x + 2)}^{\frac{1}{3}})$ .

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

Step 15.7

Separate fractions.

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

Step 15.8

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

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

Step 15.9

Combine $\frac{1}{3 {(x + 2)}^{\frac{2}{3}}}$ and $\tan ({(x + 2)}^{\frac{1}{3}})$ .

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