Calculus Examples

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

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \csc^{2} (x)$ with respect to $x$ is $- \frac{d}{d x} [\csc^{2} (x)]$ .

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

Step 2

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 2.1

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

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

Step 2.2

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

$- (2 u \frac{d}{d x} [\csc (x)])$

Step 2.3

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

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

Step 3

Multiply $2$ by $- 1$ .

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

Step 4

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

$- 2 \csc (x) (- \csc (x) \cot (x))$

Step 5

Multiply $- 1$ by $- 2$ .

$2 \csc (x) (\csc (x) \cot (x))$

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Add $1$ and $1$ .

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