What is the derivative of (cot(x-csc(x))^{-1}) ?

The derivative of (cot(x-csc(x))^{-1}) is -(csc(x))/(cot(x)-csc(x))

What is the first derivative of (cot(x-csc(x))^{-1}) ?

The first derivative of (cot(x-csc(x))^{-1}) is -(csc(x))/(cot(x)-csc(x))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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What is the derivative of (cot(x-csc(x))^{-1}) ?
The derivative of (cot(x-csc(x))^{-1}) is -(csc(x))/(cot(x)-csc(x))
What is the first derivative of (cot(x-csc(x))^{-1}) ?
The first derivative of (cot(x-csc(x))^{-1}) is -(csc(x))/(cot(x)-csc(x))