Calculus Examples

$1 - \cos (9 x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $1 - \cos (9 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (9 x)]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (9 x)]$

Step 1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \cos (9 x)]$

Step 2

Evaluate $\frac{d}{d x} [- \cos (9 x)]$ .

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (9 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (9 x)]$ .

$0 - \frac{d}{d x} [\cos (9 x)]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 9 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $9 x$ .

$0 - (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [9 x])$

Step 2.2.2

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

$0 - (- \sin (u) \frac{d}{d x} [9 x])$

Step 2.2.3

Replace all occurrences of $u$ with $9 x$ .

$0 - (- \sin (9 x) \frac{d}{d x} [9 x])$

Step 2.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$0 - (- \sin (9 x) (9 \frac{d}{d x} [x]))$

Step 2.4

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

$0 - (- \sin (9 x) (9 \cdot 1))$

Step 2.5

Multiply $9$ by $1$ .

$0 - (- \sin (9 x) \cdot 9)$

Step 2.6

Multiply $9$ by $- 1$ .

$0 - (- 9 \sin (9 x))$

Step 2.7

Multiply $- 9$ by $- 1$ .

$0 + 9 \sin (9 x)$

Step 3

Add $0$ and $9 \sin (9 x)$ .

$9 \sin (9 x)$