Find dy/dx y=(csc(x)+cot(x))^-1

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

Differentiate both sides of the equation.

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

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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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 $g (x) = \csc (x) + \cot (x)$ .

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

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

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

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

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

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

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

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

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

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

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

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

Reform the equation by setting the left side equal to the right side.

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

Simplify.

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Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y' = - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc (x) \cot (x) - \csc^{2} (x))$

Apply the distributive property.

$y' = - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc (x) \cot (x)) - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc^{2} (x))$

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

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Multiply $- 1$ by $- 1$ .

$y' = 1 (\frac{1}{{(\csc (x) + \cot (x))}^{2}}) \cdot (\csc (x) \cot (x)) - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc^{2} (x))$

Multiply $\frac{1}{{(\csc (x) + \cot (x))}^{2}}$ by $1$ .

$y' = \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (\csc (x) \cot (x)) - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc^{2} (x))$

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

$y' = \frac{\csc (x)}{{(\csc (x) + \cot (x))}^{2}} \cdot \cot (x) - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc^{2} (x))$

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

$y' = \frac{\csc (x) \cot (x)}{{(\csc (x) + \cot (x))}^{2}} - \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot (- \csc^{2} (x))$

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

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Multiply $- 1$ by $- 1$ .

$y' = \frac{\csc (x) \cot (x)}{{(\csc (x) + \cot (x))}^{2}} + 1 (\frac{1}{{(\csc (x) + \cot (x))}^{2}}) \cdot \csc^{2} (x)$

Multiply $\frac{1}{{(\csc (x) + \cot (x))}^{2}}$ by $1$ .

$y' = \frac{\csc (x) \cot (x)}{{(\csc (x) + \cot (x))}^{2}} + \frac{1}{{(\csc (x) + \cot (x))}^{2}} \cdot \csc^{2} (x)$

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

$y' = \frac{\csc (x) \cot (x)}{{(\csc (x) + \cot (x))}^{2}} + \frac{\csc^{2} (x)}{{(\csc (x) + \cot (x))}^{2}}$

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{\csc (x) \cot (x)}{{(\csc (x) + \cot (x))}^{2}} + \frac{\csc^{2} (x)}{{(\csc (x) + \cot (x))}^{2}}$

Find dy/dx y=(csc(x)+cot(x))^-1

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