Mister Exam

Derivative of log(cos(2x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(cos(2*x))
$$\log{\left(\cos{\left(2 x \right)} \right)}$$
d                
--(log(cos(2*x)))
dx               
$$\frac{d}{d x} \log{\left(\cos{\left(2 x \right)} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

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

    1. Let .

    2. The derivative of cosine is negative sine:

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

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
-2*sin(2*x)
-----------
  cos(2*x) 
$$- \frac{2 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$
The second derivative [src]
   /       2     \
   |    sin (2*x)|
-4*|1 + ---------|
   |       2     |
   \    cos (2*x)/
$$- 4 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right)$$
The third derivative [src]
    /       2     \         
    |    sin (2*x)|         
-16*|1 + ---------|*sin(2*x)
    |       2     |         
    \    cos (2*x)/         
----------------------------
          cos(2*x)          
$$- \frac{16 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$
The graph
Derivative of log(cos(2x))