Calculus Examples

\frac{\sin (x)}{x}

$\frac{\sin (x)}{x}$

Step 1

Differentiate using the Quotient Rule which states that

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

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

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = \sin (x)

$f (x) = \sin (x)$ and

g (x) = x

$g (x) = x$ .

\frac{x \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [x]}{x^{2}}

$\frac{x \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [x]}{x^{2}}$

Step 2

The derivative of

\sin (x)

$\sin (x)$ with respect to

x

$x$ is

\cos (x)

$\cos (x)$ .

\frac{x \cos (x) - \sin (x) \frac{d}{d x} [x]}{x^{2}}

$\frac{x \cos (x) - \sin (x) \frac{d}{d x} [x]}{x^{2}}$

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}]

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

\frac{x \cos (x) - \sin (x) \cdot 1}{x^{2}}

$\frac{x \cos (x) - \sin (x) \cdot 1}{x^{2}}$

Step 3.2

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{x \cos (x) - \sin (x)}{x^{2}}

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

\frac{x \cos (x) - \sin (x)}{x^{2}}

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

\frac{\sin x}{x}

$\frac{\sin x}{x}$

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