Calculus Examples

$x^{6} \cos (x)$

Step 1

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 $f (x) = x^{6}$ and $g (x) = \cos (x)$ .

$x^{6} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{6}]$

Step 2

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

$x^{6} (- \sin (x)) + \cos (x) \frac{d}{d x} [x^{6}]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$x^{6} (- \sin (x)) + \cos (x) (6 x^{5})$

Step 3.2

Reorder terms.

$- x^{6} \sin (x) + 6 x^{5} \cos (x)$