Mister Exam

Derivative of ln(cos(x))

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

The graph:

from to

Piecewise:

The solution

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

  2. The derivative of is .

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

    1. The derivative of cosine is negative sine:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

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