Calculus Examples

$\frac{17}{2} - \frac{3}{2} \cos (\frac{π t}{6})$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $\frac{17}{2} - \frac{3}{2} \cos (\frac{π t}{6})$ with respect to $t$ is $\frac{d}{d t} [\frac{17}{2}] + \frac{d}{d t} [- \frac{3}{2} \cos (\frac{π t}{6})]$ .

$\frac{d}{d t} [\frac{17}{2}] + \frac{d}{d t} [- \frac{3}{2} \cos (\frac{π t}{6})]$

Step 1.2

Since $\frac{17}{2}$ is constant with respect to $t$ , the derivative of $\frac{17}{2}$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- \frac{3}{2} \cos (\frac{π t}{6})]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{3}{2} \cos (\frac{π t}{6})]$ .

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Step 2.1

Combine $\cos (\frac{π t}{6})$ and $\frac{3}{2}$ .

$0 + \frac{d}{d t} [- \frac{\cos (\frac{π t}{6}) \cdot 3}{2}]$

Step 2.2

Move $3$ to the left of $\cos (\frac{π t}{6})$ .

$0 + \frac{d}{d t} [- \frac{3 \cdot \cos (\frac{π t}{6})}{2}]$

Step 2.3

Since $- \frac{3}{2}$ is constant with respect to $t$ , the derivative of $- \frac{3 \cos (\frac{π t}{6})}{2}$ with respect to $t$ is $- \frac{3}{2} \frac{d}{d t} [\cos (\frac{π t}{6})]$ .

$0 - \frac{3}{2} \frac{d}{d t} [\cos (\frac{π t}{6})]$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = \frac{π t}{6}$ .

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Step 2.4.1

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

$0 - \frac{3}{2} (\frac{d}{d u} [\cos (u)] \frac{d}{d t} [\frac{π t}{6}])$

Step 2.4.2

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

$0 - \frac{3}{2} (- \sin (u) \frac{d}{d t} [\frac{π t}{6}])$

Step 2.4.3

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

$0 - \frac{3}{2} (- \sin (\frac{π t}{6}) \frac{d}{d t} [\frac{π t}{6}])$

Step 2.5

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

$0 - \frac{3}{2} (- \sin (\frac{π t}{6}) (\frac{π}{6} \frac{d}{d t} [t]))$

Step 2.6

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

$0 - \frac{3}{2} (- \sin (\frac{π t}{6}) (\frac{π}{6} \cdot 1))$

Step 2.7

Multiply $\frac{π}{6}$ by $1$ .

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

Step 2.8

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

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

Step 2.9

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{3}{2}) \frac{π \sin (\frac{π t}{6})}{6}$

Step 2.10

Multiply $\frac{3}{2}$ by $1$ .

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

Step 2.11

Multiply $\frac{3}{2}$ by $\frac{π \sin (\frac{π t}{6})}{6}$ .

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

Step 2.12

Multiply $2$ by $6$ .

$0 + \frac{3 (π \sin (\frac{π t}{6}))}{12}$

Step 2.13

Cancel the common factors.

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Step 2.13.1

Factor $3$ out of $12$ .

$0 + \frac{3 (π \sin (\frac{π t}{6}))}{3 \cdot 4}$

Step 2.13.2

Cancel the common factor.

$0 + \frac{3 (π \sin (\frac{π t}{6}))}{3 \cdot 4}$

Step 2.13.3

Rewrite the expression.

$0 + \frac{π \sin (\frac{π t}{6})}{4}$

Step 3

Add $0$ and $\frac{π \sin (\frac{π t}{6})}{4}$ .

$\frac{π \sin (\frac{π t}{6})}{4}$