Calculus Examples

$g (t) = 7 t^{3} \cos (t)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = 7 t^{3}$ and $g (t) = \cos (t)$ .

$7 t^{3} \frac{d}{d t} [\cos (t)] + \cos (t) \frac{d}{d t} [7 t^{3}]$

Step 2

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

$7 t^{3} (- \sin (t)) + \cos (t) \frac{d}{d t} [7 t^{3}]$

Step 3

Differentiate.

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Step 3.1

Since $7$ is constant with respect to $t$ , the derivative of $7 t^{3}$ with respect to $t$ is $7 \frac{d}{d t} [t^{3}]$ .

$7 t^{3} (- \sin (t)) + \cos (t) (7 \frac{d}{d t} [t^{3}])$

Step 3.2

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

$7 t^{3} (- \sin (t)) + \cos (t) (7 (3 t^{2}))$

Step 3.3

Multiply $3$ by $7$ .

$7 t^{3} (- \sin (t)) + \cos (t) (21 t^{2})$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $7$ .

$- 7 t^{3} \sin (t) + \cos (t) (21 t^{2})$

Step 4.2

Reorder terms.

$- 7 t^{3} \sin (t) + 21 t^{2} \cos (t)$