Calculus Examples

$\cos (t) + t \sin (t)$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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 $f (t) = t$ and $g (t) = \sin (t)$ .

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

Step 3.2

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

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

Step 3.3

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

$- \sin (t) + t \cos (t) + \sin (t) \cdot 1$

Step 3.4

Multiply $\sin (t)$ by $1$ .

$- \sin (t) + t \cos (t) + \sin (t)$

Step 4

Combine terms.

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Step 4.1

Add $- \sin (t)$ and $\sin (t)$ .

$t \cos (t) + 0$

Step 4.2

Add $t \cos (t)$ and $0$ .

$t \cos (t)$