Calculus Examples

$x (t) = 8 \cos (t)$

Step 1

Find the first derivative.

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Since $8$ is constant with respect to $t$ , the derivative of $8 \cos (t)$ with respect to $t$ is $8 \frac{d}{d t} [\cos (t)]$ .

$f' (t) = 8 \frac{d}{d t} (\cos (t))$

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

$f' (t) = 8 (- \sin (t))$

Multiply $- 1$ by $8$ .

$f' (t) = - 8 \sin (t)$

Step 2

Find the second derivative.

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Since $- 8$ is constant with respect to $t$ , the derivative of $- 8 \sin (t)$ with respect to $t$ is $- 8 \frac{d}{d t} [\sin (t)]$ .

$f'' (t) = - 8 \frac{d}{d t} \sin (t)$

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

$f'' (t) = - 8 \cos (t)$

Step 3

The second derivative of $x (t)$ with respect to $t$ is $- 8 \cos (t)$ .

$- 8 \cos (t)$