Calculus Examples

$t^{4} \sin (t)$

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 $f (t) = t^{4}$ and $g (t) = \sin (t)$ .

$t^{4} \frac{d}{d t} [\sin (t)] + \sin (t) \frac{d}{d t} [t^{4}]$

Step 2

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

$t^{4} \cos (t) + \sin (t) \frac{d}{d t} [t^{4}]$

Step 3

Differentiate using the Power Rule.

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Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$t^{4} \cos (t) + \sin (t) (4 t^{3})$

Reorder terms.

$t^{4} \cos (t) + 4 t^{3} \sin (t)$