Calculus Examples

$f (x) = \ln (\cot (x) - \csc (x))$

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 1.3

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

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

Step 2

By the Sum Rule, the derivative of $\cot (x) - \csc (x)$ with respect to $x$ is $\frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- \csc (x)]$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Reorder the factors of $\frac{1}{\cot (x) - \csc (x)} (- \csc^{2} (x) + \csc (x) \cot (x))$ .

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine $\frac{1}{\cot (x) - \csc (x)}$ and $\csc^{2} (x)$ .

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

Step 7.4

Multiply $\csc (x) \cot (x) \frac{1}{\cot (x) - \csc (x)}$ .

Tap for more steps...

Step 7.4.1

Combine $\frac{1}{\cot (x) - \csc (x)}$ and $\csc (x)$ .

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

Step 7.4.2

Combine $\frac{\csc (x)}{\cot (x) - \csc (x)}$ and $\cot (x)$ .

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

Step 7.5

Combine the numerators over the common denominator.

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

Step 7.6

Factor $\csc (x)$ out of $- \csc^{2} (x) + \csc (x) \cot (x)$ .

Tap for more steps...

Step 7.6.1

Factor $\csc (x)$ out of $- \csc^{2} (x)$ .

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

Step 7.6.2

Factor $\csc (x)$ out of $\csc (x) \cot (x)$ .

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

Step 7.6.3

Factor $\csc (x)$ out of $\csc (x) (- \csc (x)) + \csc (x) (\cot (x))$ .

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

Step 7.7

Factor $- 1$ out of $- \csc (x)$ .

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

Step 7.8

Factor $- 1$ out of $\cot (x)$ .

$\frac{\csc (x) (- (\csc (x)) - 1 (- \cot (x)))}{\cot (x) - \csc (x)}$

Step 7.9

Factor $- 1$ out of $- (\csc (x)) - 1 (- \cot (x))$ .

$\frac{\csc (x) (- (\csc (x) - \cot (x)))}{\cot (x) - \csc (x)}$

Step 7.10

Rewrite $- (\csc (x) - \cot (x))$ as $- 1 (\csc (x) - \cot (x))$ .

$\frac{\csc (x) (- 1 (\csc (x) - \cot (x)))}{\cot (x) - \csc (x)}$

Step 7.11

Move the negative in front of the fraction.

$- \frac{\csc (x) (\csc (x) - \cot (x))}{\cot (x) - \csc (x)}$