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Popular >

derivative of ln((cos(x)/(sin(x))))

Solution

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

Solution

- 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)

Popular Examples

laplacetransform-e^t+1/2 sin(t)+1/2 cos(t)laplace transform

- e^{t} + \frac{1}{2} sin (t) + \frac{1}{2} cos (t)

integral of cos(ax+b)

\int cos (a x + b) d x

integral from-1 to 3 of 6x(x-1)

\int_{- 1}^{3} 6 x (x - 1) d x

integral of csch^2(x)

\int csch^{2} (x) d x

limit as x approaches-3 of 3x

x \to - 3 lim (3 x)

Frequently Asked Questions (FAQ)

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)