Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{2}{\sqrt[3]{x}} + 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{\sqrt[3]{x}}] + \frac{d}{d x} [3 \cos (x)]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{2}{\sqrt[3]{x}}]$ .

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Step 2.1

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

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

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{\frac{1}{3}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$ .

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

Step 2.3

Rewrite $\frac{1}{x^{\frac{1}{3}}}$ as ${(x^{\frac{1}{3}})}^{- 1}$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

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

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

Step 2.6

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 2}$ .

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

Move the negative in front of the fraction.

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $3$ .

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

Step 2.10.2

Subtract $3$ from $1$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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Step 2.14.1

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

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

Step 2.14.2

Combine the numerators over the common denominator.

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

Step 2.14.3

Subtract $2$ from $- 2$ .

$2 (- \frac{x^{\frac{- 4}{3}}}{3}) + \frac{d}{d x} [3 \cos (x)]$

Step 2.14.4

Move the negative in front of the fraction.

$2 (- \frac{x^{- \frac{4}{3}}}{3}) + \frac{d}{d x} [3 \cos (x)]$

Step 2.15

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (- \frac{1}{3 x^{\frac{4}{3}}}) + \frac{d}{d x} [3 \cos (x)]$

Step 2.16

Multiply $- 1$ by $2$ .

$- 2 \frac{1}{3 x^{\frac{4}{3}}} + \frac{d}{d x} [3 \cos (x)]$

Step 2.17

Combine $- 2$ and $\frac{1}{3 x^{\frac{4}{3}}}$ .

$\frac{- 2}{3 x^{\frac{4}{3}}} + \frac{d}{d x} [3 \cos (x)]$

Step 2.18

Move the negative in front of the fraction.

$- \frac{2}{3 x^{\frac{4}{3}}} + \frac{d}{d x} [3 \cos (x)]$

Step 3

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

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Step 3.1

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

$- \frac{2}{3 x^{\frac{4}{3}}} + 3 \frac{d}{d x} [\cos (x)]$

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $3$ .

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