Calculus Examples

$t \cos (t)$

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

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

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

Simplify the expression.

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Step 3.2.1

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

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

Step 3.2.2

Reorder terms.

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