Calculus Examples

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

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \csc^{2} (x) \cot (x)$ with respect to $x$ is $2 \frac{d}{d x} [\csc^{2} (x) \cot (x)]$ .

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

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 $f (x) = \csc^{2} (x)$ and $g (x) = \cot (x)$ .

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

Step 3

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

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

Step 4

Multiply $\csc^{2} (x)$ by $\csc^{2} (x)$ by adding the exponents.

Tap for more steps...

Step 4.1

Move $\csc^{2} (x)$ .

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

Step 4.2

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

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

Step 4.3

Add $2$ and $2$ .

$2 (\csc^{4} (x) \cdot - 1 + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Move $- 1$ to the left of $\csc^{4} (x)$ .

$2 (- 1 \cdot \csc^{4} (x) + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 5.2

Rewrite $- 1 \csc^{4} (x)$ as $- \csc^{4} (x)$ .

$2 (- \csc^{4} (x) + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \csc (x)$ .

Tap for more steps...

Step 6.1

To apply the Chain Rule, set $u$ as $\csc (x)$ .

$2 (- \csc^{4} (x) + \cot (x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\csc (x)]))$

Step 6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 (- \csc^{4} (x) + \cot (x) (2 u \frac{d}{d x} [\csc (x)]))$

Step 6.3

Replace all occurrences of $u$ with $\csc (x)$ .

$2 (- \csc^{4} (x) + \cot (x) (2 \csc (x) \frac{d}{d x} [\csc (x)]))$

Step 7

Move $2$ to the left of $\cot (x)$ .

$2 (- \csc^{4} (x) + 2 \cdot \cot (x) \csc (x) \frac{d}{d x} [\csc (x)])$

Step 8

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

$2 (- \csc^{4} (x) + 2 \cot (x) \csc (x) (- \csc (x) \cot (x)))$

Step 9

Multiply $- 1$ by $2$ .

$2 (- \csc^{4} (x) - 2 \cot (x) \csc (x) (\csc (x) \cot (x)))$

Step 10

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

$2 (- \csc^{4} (x) - 2 (\cot^{1} (x) \cot (x)) \csc (x) \csc (x))$

Step 11

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

$2 (- \csc^{4} (x) - 2 (\cot^{1} (x) \cot^{1} (x)) \csc (x) \csc (x))$

Step 12

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

$2 (- \csc^{4} (x) - 2 {\cot (x)}^{1 + 1} \csc (x) \csc (x))$

Step 13

Add $1$ and $1$ .

$2 (- \csc^{4} (x) - 2 \cot^{2} (x) \csc (x) \csc (x))$

Step 14

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

$2 (- \csc^{4} (x) - 2 \cot^{2} (x) (\csc^{1} (x) \csc (x)))$

Step 15

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

$2 (- \csc^{4} (x) - 2 \cot^{2} (x) (\csc^{1} (x) \csc^{1} (x)))$

Step 16

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

$2 (- \csc^{4} (x) - 2 \cot^{2} (x) {\csc (x)}^{1 + 1})$

Step 17

Add $1$ and $1$ .

$2 (- \csc^{4} (x) - 2 \cot^{2} (x) \csc^{2} (x))$

Step 18

Simplify.

Tap for more steps...

Step 18.1

Apply the distributive property.

$2 (- \csc^{4} (x)) + 2 (- 2 \cot^{2} (x) \csc^{2} (x))$

Step 18.2

Combine terms.

Tap for more steps...

Step 18.2.1

Multiply $- 1$ by $2$ .

$- 2 \csc^{4} (x) + 2 (- 2 \cot^{2} (x) \csc^{2} (x))$

Step 18.2.2

Multiply $- 2$ by $2$ .

$- 2 \csc^{4} (x) - 4 \cot^{2} (x) \csc^{2} (x)$

Step 18.3

Reorder terms.

$- 4 \cot^{2} (x) \csc^{2} (x) - 2 \csc^{4} (x)$