Calculus Examples

$y = 5 \cos (9 x + 1)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (9 x + 1)$ with respect to $x$ is $5 \frac{d}{d x} [\cos (9 x + 1)]$ .

$5 \frac{d}{d x} [\cos (9 x + 1)]$

Step 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 $g (x) = 9 x + 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $9 x + 1$ .

$5 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [9 x + 1])$

Step 2.2

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

$5 (- \sin (u) \frac{d}{d x} [9 x + 1])$

Step 2.3

Replace all occurrences of $u$ with $9 x + 1$ .

$5 (- \sin (9 x + 1) \frac{d}{d x} [9 x + 1])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $5$ .

$- 5 (\sin (9 x + 1) \frac{d}{d x} [9 x + 1])$

Step 3.2

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

$- 5 \sin (9 x + 1) (\frac{d}{d x} [9 x] + \frac{d}{d x} [1])$

Step 3.3

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

$- 5 \sin (9 x + 1) (9 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 3.4

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

$- 5 \sin (9 x + 1) (9 \cdot 1 + \frac{d}{d x} [1])$

Step 3.5

Multiply $9$ by $1$ .

$- 5 \sin (9 x + 1) (9 + \frac{d}{d x} [1])$

Step 3.6

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

$- 5 \sin (9 x + 1) (9 + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $9$ and $0$ .

$- 5 \sin (9 x + 1) \cdot 9$

Step 3.7.2

Multiply $9$ by $- 5$ .

$- 45 \sin (9 x + 1)$