Calculus Examples

$500 t + \frac{4200}{π} \cdot \sin (\frac{π t}{12})$

Step 1

By the Sum Rule, the derivative of $500 t + \frac{4200}{π} \cdot \sin (\frac{π t}{12})$ with respect to $t$ is $\frac{d}{d t} [500 t] + \frac{d}{d t} [\frac{4200}{π} \cdot \sin (\frac{π t}{12})]$ .

$\frac{d}{d t} [500 t] + \frac{d}{d t} [\frac{4200}{π} \cdot \sin (\frac{π t}{12})]$

Step 2

Evaluate $\frac{d}{d t} [500 t]$ .

Tap for more steps...

Step 2.1

Since $500$ is constant with respect to $t$ , the derivative of $500 t$ with respect to $t$ is $500 \frac{d}{d t} [t]$ .

$500 \frac{d}{d t} [t] + \frac{d}{d t} [\frac{4200}{π} \cdot \sin (\frac{π t}{12})]$

Step 2.2

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

$500 \cdot 1 + \frac{d}{d t} [\frac{4200}{π} \cdot \sin (\frac{π t}{12})]$

Step 2.3

Multiply $500$ by $1$ .

$500 + \frac{d}{d t} [\frac{4200}{π} \cdot \sin (\frac{π t}{12})]$

Step 3

Evaluate $\frac{d}{d t} [\frac{4200}{π} \cdot \sin (\frac{π t}{12})]$ .

Tap for more steps...

Step 3.1

Combine $\frac{4200}{π}$ and $\sin (\frac{π t}{12})$ .

$500 + \frac{d}{d t} [\frac{4200 \sin (\frac{π t}{12})}{π}]$

Step 3.2

Since $\frac{4200}{π}$ is constant with respect to $t$ , the derivative of $\frac{4200 \sin (\frac{π t}{12})}{π}$ with respect to $t$ is $\frac{4200}{π} \frac{d}{d t} [\sin (\frac{π t}{12})]$ .

$500 + \frac{4200}{π} \frac{d}{d t} [\sin (\frac{π t}{12})]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \sin (t)$ and $g (t) = \frac{π t}{12}$ .

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u$ as $\frac{π t}{12}$ .

$500 + \frac{4200}{π} (\frac{d}{d u} [\sin (u)] \frac{d}{d t} [\frac{π t}{12}])$

Step 3.3.2

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

$500 + \frac{4200}{π} (\cos (u) \frac{d}{d t} [\frac{π t}{12}])$

Step 3.3.3

Replace all occurrences of $u$ with $\frac{π t}{12}$ .

$500 + \frac{4200}{π} (\cos (\frac{π t}{12}) \frac{d}{d t} [\frac{π t}{12}])$

Step 3.4

Since $\frac{π}{12}$ is constant with respect to $t$ , the derivative of $\frac{π t}{12}$ with respect to $t$ is $\frac{π}{12} \frac{d}{d t} [t]$ .

$500 + \frac{4200}{π} (\cos (\frac{π t}{12}) (\frac{π}{12} \frac{d}{d t} [t]))$

Step 3.5

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

$500 + \frac{4200}{π} (\cos (\frac{π t}{12}) (\frac{π}{12} \cdot 1))$

Step 3.6

Multiply $\frac{π}{12}$ by $1$ .

$500 + \frac{4200}{π} (\cos (\frac{π t}{12}) \frac{π}{12})$

Step 3.7

Combine $\cos (\frac{π t}{12})$ and $\frac{π}{12}$ .

$500 + \frac{4200}{π} \cdot \frac{\cos (\frac{π t}{12}) π}{12}$

Step 3.8

Multiply $\frac{4200}{π}$ by $\frac{\cos (\frac{π t}{12}) π}{12}$ .

$500 + \frac{4200 (\cos (\frac{π t}{12}) π)}{π \cdot 12}$

Step 3.9

Move $12$ to the left of $π$ .

$500 + \frac{4200 \cos (\frac{π t}{12}) π}{12 \cdot π}$

Step 3.10

Cancel the common factor of $4200$ and $12$ .

Tap for more steps...

Step 3.10.1

Factor $12$ out of $4200 \cos (\frac{π t}{12}) π$ .

$500 + \frac{12 (350 \cos (\frac{π t}{12}) π)}{12 π}$

Step 3.10.2

Cancel the common factors.

Tap for more steps...

Step 3.10.2.1

Factor $12$ out of $12 π$ .

$500 + \frac{12 (350 \cos (\frac{π t}{12}) π)}{12 (π)}$

Step 3.10.2.2

Cancel the common factor.

$500 + \frac{12 (350 \cos (\frac{π t}{12}) π)}{12 π}$

Step 3.10.2.3

Rewrite the expression.

$500 + \frac{350 \cos (\frac{π t}{12}) π}{π}$

Step 3.11

Cancel the common factor of $π$ .

Tap for more steps...

Step 3.11.1

Cancel the common factor.

$500 + \frac{350 \cos (\frac{π t}{12}) π}{π}$

Step 3.11.2

Divide $350 \cos (\frac{π t}{12})$ by $1$ .

$500 + 350 \cos (\frac{π t}{12})$