Calculus Examples

$x^{4} \sin (x)$

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 $f (x) = x^{4}$ and $g (x) = \sin (x)$ .

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

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

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

Differentiate using the Power Rule.

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Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Simplify the expression.

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Remove parentheses around $4 x^{3}$ .

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

Move $4$ to the left of the expression $\sin (x) \cdot 4$ .

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

Multiply $4$ by $\sin (x)$ .

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

Reorder terms.

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

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