Calculus Examples

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

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \csc (x) \cot (x)$ with respect to $x$ is $- \frac{d}{d x} [\csc (x) \cot (x)]$ .

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

Step 2

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 $f (x) = \csc (x)$ and $g (x) = \cot (x)$ .

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

Step 3

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

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

Step 4

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

$- (- (\csc^{1} (x) \csc^{2} (x)) + \cot (x) \frac{d}{d x} [\csc (x)])$

Step 5

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

$- (- {\csc (x)}^{1 + 2} + \cot (x) \frac{d}{d x} [\csc (x)])$

Step 6

Add $1$ and $2$ .

$- (- \csc^{3} (x) + \cot (x) \frac{d}{d x} [\csc (x)])$

Step 7

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

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

Step 8

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

$- (- \csc^{3} (x) - \csc (x) (\cot^{1} (x) \cot (x)))$

Step 9

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

$- (- \csc^{3} (x) - \csc (x) (\cot^{1} (x) \cot^{1} (x)))$

Step 10

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

$- (- \csc^{3} (x) - \csc (x) {\cot (x)}^{1 + 1})$

Step 11

Add $1$ and $1$ .

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

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Combine terms.

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Step 12.2.1

Multiply $- 1$ by $- 1$ .

$1 \csc^{3} (x) - (- \csc (x) \cot^{2} (x))$

Step 12.2.2

Multiply $\csc^{3} (x)$ by $1$ .

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

Step 12.2.3

Multiply $- 1$ by $- 1$ .

$\csc^{3} (x) + 1 (\csc (x) \cot^{2} (x))$

Step 12.2.4

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

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

Step 12.3

Reorder terms.

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