Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$\frac{\sin (x) \cdot 1 - x \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}$

Step 2.2

Multiply $\sin (x)$ by $1$ .

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

Step 3

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

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

Step 4

Reorder terms.

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