Calculus Examples

$20 + 9 \cos (\frac{π}{6} x)$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $20 + 9 \cos (\frac{π}{6} x)$ with respect to $x$ is $\frac{d}{d x} [20] + \frac{d}{d x} [9 \cos (\frac{π}{6} x)]$ .

$\frac{d}{d x} [20] + \frac{d}{d x} [9 \cos (\frac{π}{6} x)]$

Step 1.2

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [9 \cos (\frac{π}{6} x)]$

Step 2

Evaluate $\frac{d}{d x} [9 \cos (\frac{π}{6} x)]$ .

Tap for more steps...

Step 2.1

Combine $\frac{π}{6}$ and $x$ .

$0 + \frac{d}{d x} [9 \cos (\frac{π x}{6})]$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 \cos (\frac{π x}{6})$ with respect to $x$ is $9 \frac{d}{d x} [\cos (\frac{π x}{6})]$ .

$0 + 9 \frac{d}{d x} [\cos (\frac{π x}{6})]$

Step 2.3

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 $g (x) = \frac{π x}{6}$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $\frac{π x}{6}$ .

$0 + 9 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{π x}{6}])$

Step 2.3.2

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

$0 + 9 (- \sin (u) \frac{d}{d x} [\frac{π x}{6}])$

Step 2.3.3

Replace all occurrences of $u$ with $\frac{π x}{6}$ .

$0 + 9 (- \sin (\frac{π x}{6}) \frac{d}{d x} [\frac{π x}{6}])$

Step 2.4

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

$0 + 9 (- \sin (\frac{π x}{6}) (\frac{π}{6} \frac{d}{d x} [x]))$

Step 2.5

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

$0 + 9 (- \sin (\frac{π x}{6}) (\frac{π}{6} \cdot 1))$

Step 2.6

Multiply $\frac{π}{6}$ by $1$ .

$0 + 9 (- \sin (\frac{π x}{6}) \frac{π}{6})$

Step 2.7

Combine $\frac{π}{6}$ and $\sin (\frac{π x}{6})$ .

$0 + 9 (- \frac{π \sin (\frac{π x}{6})}{6})$

Step 2.8

Multiply $- 1$ by $9$ .

$0 - 9 \frac{π \sin (\frac{π x}{6})}{6}$

Step 2.9

Combine $- 9$ and $\frac{π \sin (\frac{π x}{6})}{6}$ .

$0 + \frac{- 9 (π \sin (\frac{π x}{6}))}{6}$

Step 2.10

Cancel the common factor of $- 9$ and $6$ .

Tap for more steps...

Step 2.10.1

Factor $3$ out of $- 9 π \sin (\frac{π x}{6})$ .

$0 + \frac{3 (- 3 π \sin (\frac{π x}{6}))}{6}$

Step 2.10.2

Cancel the common factors.

Tap for more steps...

Step 2.10.2.1

Factor $3$ out of $6$ .

$0 + \frac{3 (- 3 π \sin (\frac{π x}{6}))}{3 (2)}$

Step 2.10.2.2

Cancel the common factor.

$0 + \frac{3 (- 3 π \sin (\frac{π x}{6}))}{3 \cdot 2}$

Step 2.10.2.3

Rewrite the expression.

$0 + \frac{- 3 π \sin (\frac{π x}{6})}{2}$

Step 2.11

Move the negative in front of the fraction.

$0 - \frac{3 π \sin (\frac{π x}{6})}{2}$

Step 3

Subtract $\frac{3 π \sin (\frac{π x}{6})}{2}$ from $0$ .

$- \frac{3 π \sin (\frac{π x}{6})}{2}$