Mister Exam

Derivative of log(sin(x))

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

The graph:

from to

Piecewise:

The solution

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

  2. The derivative of is .

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

    1. The derivative of sine is cosine:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

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