Calculus Examples

$s = 7 \sin (t) + 14 \cos (t)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $7 \sin (t) + 14 \cos (t)$ with respect to $t$ is $\frac{d}{d t} [7 \sin (t)] + \frac{d}{d t} [14 \cos (t)]$ .

$\frac{d}{d t} [7 \sin (t)] + \frac{d}{d t} [14 \cos (t)]$

Step 1.2

Evaluate $\frac{d}{d t} [7 \sin (t)]$ .

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Step 1.2.1

Since $7$ is constant with respect to $t$ , the derivative of $7 \sin (t)$ with respect to $t$ is $7 \frac{d}{d t} [\sin (t)]$ .

$7 \frac{d}{d t} [\sin (t)] + \frac{d}{d t} [14 \cos (t)]$

Step 1.2.2

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

$7 \cos (t) + \frac{d}{d t} [14 \cos (t)]$

Step 1.3

Evaluate $\frac{d}{d t} [14 \cos (t)]$ .

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Step 1.3.1

Since $14$ is constant with respect to $t$ , the derivative of $14 \cos (t)$ with respect to $t$ is $14 \frac{d}{d t} [\cos (t)]$ .

$7 \cos (t) + 14 \frac{d}{d t} [\cos (t)]$

Step 1.3.2

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

$7 \cos (t) + 14 (- \sin (t))$

Step 1.3.3

Multiply $- 1$ by $14$ .

$f' (t) = 7 \cos (t) - 14 \sin (t)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $7 \cos (t) - 14 \sin (t)$ with respect to $t$ is $\frac{d}{d t} [7 \cos (t)] + \frac{d}{d t} [- 14 \sin (t)]$ .

$\frac{d}{d t} [7 \cos (t)] + \frac{d}{d t} [- 14 \sin (t)]$

Step 2.2

Evaluate $\frac{d}{d t} [7 \cos (t)]$ .

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Step 2.2.1

Since $7$ is constant with respect to $t$ , the derivative of $7 \cos (t)$ with respect to $t$ is $7 \frac{d}{d t} [\cos (t)]$ .

$7 \frac{d}{d t} [\cos (t)] + \frac{d}{d t} [- 14 \sin (t)]$

Step 2.2.2

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

$7 (- \sin (t)) + \frac{d}{d t} [- 14 \sin (t)]$

Step 2.2.3

Multiply $- 1$ by $7$ .

$- 7 \sin (t) + \frac{d}{d t} [- 14 \sin (t)]$

Step 2.3

Evaluate $\frac{d}{d t} [- 14 \sin (t)]$ .

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Step 2.3.1

Since $- 14$ is constant with respect to $t$ , the derivative of $- 14 \sin (t)$ with respect to $t$ is $- 14 \frac{d}{d t} [\sin (t)]$ .

$- 7 \sin (t) - 14 \frac{d}{d t} [\sin (t)]$

Step 2.3.2

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

$f'' (t) = - 7 \sin (t) - 14 \cos (t)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- 7 \sin (t) - 14 \cos (t)$ with respect to $t$ is $\frac{d}{d t} [- 7 \sin (t)] + \frac{d}{d t} [- 14 \cos (t)]$ .

$\frac{d}{d t} [- 7 \sin (t)] + \frac{d}{d t} [- 14 \cos (t)]$

Step 3.2

Evaluate $\frac{d}{d t} [- 7 \sin (t)]$ .

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Step 3.2.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 \sin (t)$ with respect to $t$ is $- 7 \frac{d}{d t} [\sin (t)]$ .

$- 7 \frac{d}{d t} [\sin (t)] + \frac{d}{d t} [- 14 \cos (t)]$

Step 3.2.2

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

$- 7 \cos (t) + \frac{d}{d t} [- 14 \cos (t)]$

Step 3.3

Evaluate $\frac{d}{d t} [- 14 \cos (t)]$ .

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Step 3.3.1

Since $- 14$ is constant with respect to $t$ , the derivative of $- 14 \cos (t)$ with respect to $t$ is $- 14 \frac{d}{d t} [\cos (t)]$ .

$- 7 \cos (t) - 14 \frac{d}{d t} [\cos (t)]$

Step 3.3.2

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

$- 7 \cos (t) - 14 (- \sin (t))$

Step 3.3.3

Multiply $- 1$ by $- 14$ .

$f''' (t) = - 7 \cos (t) + 14 \sin (t)$