Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $- 1$ by $3$ .

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

Step 3

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

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Step 3.1

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

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

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 $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 3.3

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

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

Step 3.4

Multiply $- 1$ by $3$ .

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

Step 3.5

Multiply $- 3$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Factor $3 \sin (x)$ out of $3 \cos^{2} (x) \sin (x) - 3 \sin (x)$ .

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Step 4.2.1

Factor $3 \sin (x)$ out of $3 \cos^{2} (x) \sin (x)$ .

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

Step 4.2.2

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

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

Step 4.2.3

Factor $3 \sin (x)$ out of $3 \sin (x) (\cos^{2} (x)) + 3 \sin (x) (- 1)$ .

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

Step 4.3

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 4.4

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.5

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 4.6

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 4.7

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 4.8

Apply pythagorean identity.

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

Step 4.9

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 4.9.1

Move $\sin^{2} (x)$ .

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

Step 4.9.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 4.9.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.9.2.2

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

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

Step 4.9.3

Add $2$ and $1$ .

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

Step 4.10

Multiply $- 1$ by $3$ .

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