Calculus Examples

$f (x) = \cos (x) - \frac{3}{x^{3}}$

Step 1

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [- \frac{3}{x^{3}}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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 $g (x) = x^{3}$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Multiply $3$ by $- 2$ .

$- \sin (x) - 3 (- x^{- 6} (3 x^{2}))$

Step 3.6

Multiply $3$ by $- 1$ .

$- \sin (x) - 3 (- 3 x^{- 6} x^{2})$

Step 3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.7.1

Move $x^{2}$ .

$- \sin (x) - 3 (- 3 (x^{2} x^{- 6}))$

Step 3.7.2

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

$- \sin (x) - 3 (- 3 x^{2 - 6})$

Step 3.7.3

Subtract $6$ from $2$ .

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

Step 3.8

Multiply $- 3$ by $- 3$ .

$- \sin (x) + 9 x^{- 4}$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \sin (x) + 9 \frac{1}{x^{4}}$

Step 4.2

Combine $9$ and $\frac{1}{x^{4}}$ .

$- \sin (x) + \frac{9}{x^{4}}$

Step 4.3

Reorder terms.

$\frac{9}{x^{4}} - \sin (x)$