Calculus Examples

$h (t) = 6 \cos (\frac{π}{6} t) + 8$

Step 1

By the Sum Rule, the derivative of $6 \cos (\frac{π}{6} t) + 8$ with respect to $t$ is $\frac{d}{d t} [6 \cos (\frac{π}{6} t)] + \frac{d}{d t} [8]$ .

$\frac{d}{d t} [6 \cos (\frac{π}{6} t)] + \frac{d}{d t} [8]$

Step 2

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

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Step 2.1

Combine $\frac{π}{6}$ and $t$ .

$\frac{d}{d t} [6 \cos (\frac{π t}{6})] + \frac{d}{d t} [8]$

Step 2.2

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

$6 \frac{d}{d t} [\cos (\frac{π t}{6})] + \frac{d}{d t} [8]$

Step 2.3

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 2.3.1

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

$6 (\frac{d}{d u} [\cos (u)] \frac{d}{d t} [\frac{π t}{6}]) + \frac{d}{d t} [8]$

Step 2.3.2

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

$6 (- \sin (u) \frac{d}{d t} [\frac{π t}{6}]) + \frac{d}{d t} [8]$

Step 2.3.3

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

$6 (- \sin (\frac{π t}{6}) \frac{d}{d t} [\frac{π t}{6}]) + \frac{d}{d t} [8]$

Step 2.4

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

$6 (- \sin (\frac{π t}{6}) (\frac{π}{6} \frac{d}{d t} [t])) + \frac{d}{d t} [8]$

Step 2.5

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

$6 (- \sin (\frac{π t}{6}) (\frac{π}{6} \cdot 1)) + \frac{d}{d t} [8]$

Step 2.6

Multiply $\frac{π}{6}$ by $1$ .

$6 (- \sin (\frac{π t}{6}) \frac{π}{6}) + \frac{d}{d t} [8]$

Step 2.7

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

$6 (- \frac{π \sin (\frac{π t}{6})}{6}) + \frac{d}{d t} [8]$

Step 2.8

Multiply $- 1$ by $6$ .

$- 6 \frac{π \sin (\frac{π t}{6})}{6} + \frac{d}{d t} [8]$

Step 2.9

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

$\frac{- 6 (π \sin (\frac{π t}{6}))}{6} + \frac{d}{d t} [8]$

Step 2.10

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

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Step 2.10.1

Factor $6$ out of $- 6 π \sin (\frac{π t}{6})$ .

$\frac{6 (- π \sin (\frac{π t}{6}))}{6} + \frac{d}{d t} [8]$

Step 2.10.2

Cancel the common factors.

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Step 2.10.2.1

Factor $6$ out of $6$ .

$\frac{6 (- π \sin (\frac{π t}{6}))}{6 (1)} + \frac{d}{d t} [8]$

Step 2.10.2.2

Cancel the common factor.

$\frac{6 (- π \sin (\frac{π t}{6}))}{6 \cdot 1} + \frac{d}{d t} [8]$

Step 2.10.2.3

Rewrite the expression.

$\frac{- π \sin (\frac{π t}{6})}{1} + \frac{d}{d t} [8]$

Step 2.10.2.4

Divide $- π \sin (\frac{π t}{6})$ by $1$ .

$- π \sin (\frac{π t}{6}) + \frac{d}{d t} [8]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $8$ is constant with respect to $t$ , the derivative of $8$ with respect to $t$ is $0$ .

$- π \sin (\frac{π t}{6}) + 0$

Step 3.2

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

$- π \sin (\frac{π t}{6})$