Calculus Examples

$s = 3 \cos (t^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (3 \cos (t^{3}))$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

$3 \frac{d}{d t} [\cos (t^{3})]$

Step 3.2

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 $g (t) = t^{3}$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $t^{3}$ .

$3 (\frac{d}{d u} [\cos (u)] \frac{d}{d t} [t^{3}])$

Step 3.2.2

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

$3 (- \sin (u) \frac{d}{d t} [t^{3}])$

Step 3.2.3

Replace all occurrences of $u$ with $t^{3}$ .

$3 (- \sin (t^{3}) \frac{d}{d t} [t^{3}])$

Step 3.3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.3.1

Multiply $- 1$ by $3$ .

$- 3 (\sin (t^{3}) \frac{d}{d t} [t^{3}])$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$- 3 \sin (t^{3}) (3 t^{2})$

Step 3.3.3

Simplify the expression.

Tap for more steps...

Step 3.3.3.1

Multiply $3$ by $- 3$ .

$- 9 \sin (t^{3}) t^{2}$

Step 3.3.3.2

Reorder the factors of $- 9 \sin (t^{3}) t^{2}$ .

$- 9 t^{2} \sin (t^{3})$

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = - 9 t^{2} \sin (t^{3})$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = - 9 t^{2} \sin (t^{3})$