Calculus Examples

$\cot (x) \csc (x)$

Step 1

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 2

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 3

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

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

Step 4

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 5

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 6

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $2$ .

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

Step 11

Reorder terms.

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