Calculus Examples

y = x^{3} cos (x)

$y = x^{3} \cos (x)$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x^{3}

$f (x) = x^{3}$ and

g (x) = cos (x)

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

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

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

Step 2

The derivative of

cos (x)

$\cos (x)$ with respect to

x

$x$ is

- sin (x)

$- \sin (x)$ .

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

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

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

$n = 3$ .

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

Step 3.2

Reorder terms.

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

$y = x^{3} \cos (x)$

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