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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

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The solution

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

  1. Let u=1−cos⁡(x)u = 1 - \cos{\left(x \right)}.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(1−cos⁡(x))\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right):

    1. Differentiate 1−cos⁡(x)1 - \cos{\left(x \right)} term by term:

      1. The derivative of the constant 11 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:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        So, the result is: sin⁡(x)\sin{\left(x \right)}

      The result is: sin⁡(x)\sin{\left(x \right)}

    The result of the chain rule is:

    sin⁡(x)1−cos⁡(x)\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}

  4. Now simplify:

    −sin⁡(x)cos⁡(x)−1- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 1}

The answer is:

−sin⁡(x)cos⁡(x)−1- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 1}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  sin(x)  
----------
1 - cos(x)
sin⁡(x)−cos⁡(x)+1\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)
−cos⁡(x)+sin⁡2(x)cos⁡(x)−1cos⁡(x)−1- \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)
(1−3cos⁡(x)cos⁡(x)−1−2sin⁡2(x)(cos⁡(x)−1)2)sin⁡(x)cos⁡(x)−1\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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