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ln(1−cos(x))
The derivative of the function/ ln(1−cos(x))

Derivative of ln(1−cos(x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
log(1 - cos(x))
$$\log{\left(- \cos{\left(x \right)} + 1 \right)}$$ 
d                  
--(log(1 - cos(x)))
dx
$$\frac{d}{d x} \log{\left(- \cos{\left(x \right)} + 1 \right)}$$
Transform
Detail solution

  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. The derivative of cosine is negative sine:

        So, the result is:

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  sin(x)  
----------
1 - cos(x)
$$\frac{\sin{\left(x \right)}}{- \cos{\left(x \right)} + 1}$$
Simplify
The second derivative [src] 
 /     2              \ 
 |  sin (x)           | 
-|----------- + cos(x)| 
 \-1 + cos(x)         / 
------------------------
      -1 + cos(x)
$$- \frac{\cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{\cos{\left(x \right)} - 1}$$
Simplify
The third derivative [src] 
/                         2      \       
|      3*cos(x)      2*sin (x)   |       
|1 - ----------- - --------------|*sin(x)
|    -1 + cos(x)                2|       
\                  (-1 + cos(x)) /       
-----------------------------------------
               -1 + cos(x)
$$\frac{\left(1 - \frac{3 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1}$$
Simplify
The graph
Derivative of ln(1−cos(x))
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