Calculus Examples

$f (x) = \frac{\cot (x)}{\sin (x)}$

Step 1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 2

Rewrite $\frac{\frac{\cos (x)}{\sin (x)}}{\sin (x)}$ as a product.

$\frac{d}{d x} [\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}]$

Step 3

Simplify.

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Step 3.1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{d}{d x} [\cot (x) \frac{1}{\sin (x)}]$

Step 3.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{d}{d x} [\cot (x) \csc (x)]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cot (x)$ and $g (x) = \csc (x)$ .

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

Step 5

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)]$

Step 6

Raise $\cot (x)$ to the power of $1$ .

$- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)]$

Step 7

Raise $\cot (x)$ to the power of $1$ .

$- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) \frac{d}{d x} [\cot (x)]$

Step 8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) \frac{d}{d x} [\cot (x)]$

Step 9

Add $1$ and $1$ .

$- \csc (x) \cot^{2} (x) + \csc (x) \frac{d}{d x} [\cot (x)]$

Step 10

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x))$

Step 11

Raise $\csc (x)$ to the power of $1$ .

$- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x))$

Step 12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2}$

Step 13

Add $1$ and $2$ .

$- \csc (x) \cot^{2} (x) - \csc^{3} (x)$

Step 14

Reorder terms.

$- \cot^{2} (x) \csc (x) - \csc^{3} (x)$