What is the derivative of ln((cos(x)/(sin(x)))) ?

The derivative of ln((cos(x)/(sin(x)))) is -csc^2(x)tan(x)

What is the first derivative of ln((cos(x)/(sin(x)))) ?

The first derivative of ln((cos(x)/(sin(x)))) is -csc^2(x)tan(x)

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\frac{d}{d x} (ln (\frac{cos ( x )}{sin ( x )}))

- csc^{2} (x) tan (x)

Solution steps

\frac{d}{d x} (ln (\frac{cos ( x )}{sin ( x )}))

Simplify

ln (\frac{cos ( x )}{sin ( x )}) : ln (cot (x))

= \frac{d}{d x} (ln (cot (x)))

Apply the chain rule:

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

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

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

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

Simplify

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

= - csc^{2} (x) tan (x)

What is the derivative of ln((cos(x)/(sin(x)))) ?
The derivative of ln((cos(x)/(sin(x)))) is -csc^2(x)tan(x)
What is the first derivative of ln((cos(x)/(sin(x)))) ?
The first derivative of ln((cos(x)/(sin(x)))) is -csc^2(x)tan(x)