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\frac{d}{dx}\left(\ln\left(\frac{1+\cos\left(x\right)}{1-\cos\left(x\right)}\right)\right)

Derive the function ln((1+cos(x))/(1-1cos(x))) with respect to x

Answer

$\frac{-\left(1-\cos\left(x\right)\right)\sin\left(x\right)-\sin\left(x\right)\left(\cos\left(x\right)+1\right)}{\sin\left(x\right)^2}$

Step-by-step explanation

Problem

$\frac{d}{dx}\left(\ln\left(\frac{1+\cos\left(x\right)}{1-\cos\left(x\right)}\right)\right)$
1

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{d}{dx}\left(\frac{\cos\left(x\right)+1}{1-\cos\left(x\right)}\right)\frac{1}{\frac{\cos\left(x\right)+1}{1-\cos\left(x\right)}}$

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Answer

$\frac{-\left(1-\cos\left(x\right)\right)\sin\left(x\right)-\sin\left(x\right)\left(\cos\left(x\right)+1\right)}{\sin\left(x\right)^2}$

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$\frac{d}{dx}\left(\ln\left(\frac{1+\cos\left(x\right)}{1-\cos\left(x\right)}\right)\right)$

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 1.7 seconds