Calculus Examples

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

Step 1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \cos^{2} (x) \sin (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\cos^{2} (x) \sin (x)]$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos^{2} (x)$ and $g (x) = \sin (x)$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Add $2$ and $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Move $2$ to the left of $\sin (x)$ .

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

Step 7

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

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

Step 8

Multiply $- 1$ by $2$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Add $1$ and $1$ .

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Multiply $- 2$ by $- 3$ .

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

Step 13.3

Reorder terms.

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