Calculus Examples

$f (x) = 4 x^{3} \sin (x)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3} \sin (x)$ with respect to $x$ is $4 \frac{d}{d x} [x^{3} \sin (x)]$ .

$4 \frac{d}{d x} [x^{3} \sin (x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = \sin (x)$ .

$4 (x^{3} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{3}])$

Step 3

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

$4 (x^{3} \cos (x) + \sin (x) \frac{d}{d x} [x^{3}])$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 (x^{3} \cos (x) + \sin (x) (3 x^{2}))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 (x^{3} \cos (x)) + 4 (\sin (x) (3 x^{2}))$

Step 5.2

Multiply $3$ by $4$ .

$4 x^{3} \cos (x) + 12 \sin (x) x^{2}$

Step 5.3

Reorder terms.

$4 x^{3} \cos (x) + 12 x^{2} \sin (x)$