Calculus Examples

$3 t \cos (t)$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$3 (t (- \sin (t)) + \cos (t) \cdot 1)$

Step 4.2

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

$3 (t (- \sin (t)) + \cos (t))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$3 (t (- \sin (t))) + 3 \cos (t)$

Step 5.2

Multiply $- 1$ by $3$ .

$- 3 t \sin (t) + 3 \cos (t)$