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

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

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

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

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derivative of

(csc (x) + cot (x))^{- 1}

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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