Derivative of log(cos(x))

The solution

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log(cos(x))

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

log(cos(x))

Detail solution

Let $u = \cos{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result of the chain rule is:

$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
Now simplify:

$- \tan{\left(x \right)}$

The answer is:

$- \tan{\left(x \right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

-sin(x) 
--------
 cos(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of log(cos(x))

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The solution