Derivative of log(1+cos(x))

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log(1 + cos(x))

$$\log{\left(\cos{\left(x \right)} + 1 \right)}$$

log(1 + cos(x))

Detail solution

Let $u = \cos{\left(x \right)} + 1$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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 the constant $1$ is zero.
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $- \sin{\left(x \right)}$
The result of the chain rule is:

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

The answer is:

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

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

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Derivative of log(1+cos(x))

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The solution