Calculus Examples

$y = \frac{1}{x} \sin (x)$

Step 1

Combine $\frac{1}{x}$ and $\sin (x)$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x)$ and $g (x) = x$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Multiply $- 1$ by $1$ .

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