Calculus Examples

$- 6 \cos (x) \sin (x) - 6 \cos (x)$

Step 1

By the Sum Rule, the derivative of $- 6 \cos (x) \sin (x) - 6 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- 6 \cos (x) \sin (x)] + \frac{d}{d x} [- 6 \cos (x)]$ .

$\frac{d}{d x} [- 6 \cos (x) \sin (x)] + \frac{d}{d x} [- 6 \cos (x)]$

Step 2

Evaluate $\frac{d}{d x} [- 6 \cos (x) \sin (x)]$ .

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Step 2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \cos (x) \sin (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$- 6 \frac{d}{d x} [\cos (x) \sin (x)] + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.2

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$- 6 (\cos (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.3

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

$- 6 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.4

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

$- 6 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.5

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

$- 6 (\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.6

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

$- 6 (\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.7

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

$- 6 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.8

Add $1$ and $1$ .

$- 6 (\cos^{2} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.9

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

$- 6 (\cos^{2} (x) - (\sin^{1} (x) \sin (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.10

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

$- 6 (\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x))) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.11

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

$- 6 (\cos^{2} (x) - {\sin (x)}^{1 + 1}) + \frac{d}{d x} [- 6 \cos (x)]$

Step 2.12

Add $1$ and $1$ .

$- 6 (\cos^{2} (x) - \sin^{2} (x)) + \frac{d}{d x} [- 6 \cos (x)]$

Step 3

Evaluate $\frac{d}{d x} [- 6 \cos (x)]$ .

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Step 3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \cos (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\cos (x)]$ .

$- 6 (\cos^{2} (x) - \sin^{2} (x)) - 6 \frac{d}{d x} [\cos (x)]$

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $- 6$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $- 6$ .

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