Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = \cot (x) - \csc (x)$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

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

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