Calculus Examples

x \cos (x)

$x \cos (x)$

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

$f (x) = x$ and

g (x) = \cos (x)

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

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

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

Step 2

The derivative of

\cos (x)

$\cos (x)$ with respect to

x

$x$ is

- \sin (x)

$- \sin (x)$ .

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

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

Step 3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

x (- \sin (x)) + \cos (x) \cdot 1

$x (- \sin (x)) + \cos (x) \cdot 1$

Step 4

Simplify.

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Step 4.1

Multiply

\cos (x)

$\cos (x)$ by

1

$1$ .

x (- \sin (x)) + \cos (x)

$x (- \sin (x)) + \cos (x)$

Step 4.2

Reorder terms.

- x \sin (x) + \cos (x)

$- x \sin (x) + \cos (x)$

- x \sin (x) + \cos (x)

$- x \sin (x) + \cos (x)$

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