What is the derivative of ln(csc(x-cot(x))) ?

The derivative of ln(csc(x-cot(x))) is 1/((csc(x)-cot(x))(1+cos(x)))

What is the first derivative of ln(csc(x-cot(x))) ?

The first derivative of ln(csc(x-cot(x))) is 1/((csc(x)-cot(x))(1+cos(x)))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{csc ( x ) - cot ( x )} \cdot \frac{1}{1 + cos ( x )}

Simplify

\frac{1}{csc ( x ) - cot ( x )} \cdot \frac{1}{1 + cos ( x )} : \frac{1}{( csc ( x ) - cot ( x ) ) ( 1 + cos ( x ) )}

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

\frac{1}{x}

\int_{- 1}^{1} \frac{1}{x ^{3}} d x

\frac{5}{( s + 1 ) ( s ^{2} + 4 s + 3 )}

\frac{d h}{d t} = - h

\int 2 x^{4} d x

What is the derivative of ln(csc(x-cot(x))) ?
The derivative of ln(csc(x-cot(x))) is 1/((csc(x)-cot(x))(1+cos(x)))
What is the first derivative of ln(csc(x-cot(x))) ?
The first derivative of ln(csc(x-cot(x))) is 1/((csc(x)-cot(x))(1+cos(x)))