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 )}

\int_{0}^{2} (2 - t) t d t

\int \frac{t ^{6} + 1}{t ^{6} + t ^{3}} d t

\int 31 x^{30} d x

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\int \frac{e ^{x}}{e ^{2 x} - 9} d x

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))