Derivative of (ln(cos(x)-1))/2

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log(cos(x) - 1)
---------------
       2

$$\frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2}$$

log(cos(x) - 1)/2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos{\left(x \right)} - 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)$ :
  1. Differentiate $\cos{\left(x \right)} - 1$ term by term:
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    2. The derivative of the constant $-1$ is zero.
    The result is: $- \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 1}$
So, the result is: $- \frac{\sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

   -sin(x)    
--------------
2*(cos(x) - 1)

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

Simplify

The second derivative [src]

 /     2              \ 
 |  sin (x)           | 
-|----------- + cos(x)| 
 \-1 + cos(x)         / 
------------------------
    2*(-1 + cos(x))

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

Simplify

The third derivative [src]

 /            2                    \        
 |       2*sin (x)        3*cos(x) |        
-|-1 + -------------- + -----------|*sin(x) 
 |                  2   -1 + cos(x)|        
 \     (-1 + cos(x))               /        
--------------------------------------------
              2*(-1 + cos(x))

$$- \frac{\left(-1 + \frac{3 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{2 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)}$$

Simplify

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Derivative of (ln(cos(x)-1))/2

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The solution