Calculus Examples

$y = d \cos (t) + t^{2} \sin (t)$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of $d \cos (t) + t^{2} \sin (t)$ with respect to $t$ is $\frac{d}{d t} [d \cos (t)] + \frac{d}{d t} [t^{2} \sin (t)]$ .

$\frac{d}{d t} [d \cos (t)] + \frac{d}{d t} [t^{2} \sin (t)]$

Step 3

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

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Step 3.1

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

$d \frac{d}{d t} [\cos (t)] + \frac{d}{d t} [t^{2} \sin (t)]$

Step 3.2

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

$d (- \sin (t)) + \frac{d}{d t} [t^{2} \sin (t)]$

Step 4

Evaluate $\frac{d}{d t} [t^{2} \sin (t)]$ .

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Step 4.1

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 $f (t) = t^{2}$ and $g (t) = \sin (t)$ .

$d (- \sin (t)) + t^{2} \frac{d}{d t} [\sin (t)] + \sin (t) \frac{d}{d t} [t^{2}]$

Step 4.2

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

$d (- \sin (t)) + t^{2} \cos (t) + \sin (t) \frac{d}{d t} [t^{2}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

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

Step 5

Reorder terms.

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