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Derivative of ln^2(cos2x)

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

The graph:

from to

Piecewise:

The solution

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

  2. Apply the power rule: goes to

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

    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:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
-4*log(cos(2*x))*sin(2*x)
-------------------------
         cos(2*x)        
$$- \frac{4 \log{\left(\cos{\left(2 x \right)} \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$
The second derivative [src]
  /                    2           2                   \
  |                 sin (2*x)   sin (2*x)*log(cos(2*x))|
8*|-log(cos(2*x)) + --------- - -----------------------|
  |                    2                  2            |
  \                 cos (2*x)          cos (2*x)       /
$$8 \left(- \frac{\log{\left(\cos{\left(2 x \right)} \right)} \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} - \log{\left(\cos{\left(2 x \right)} \right)} + \frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}\right)$$
The third derivative [src]
   /                           2             2                   \         
   |                      3*sin (2*x)   2*sin (2*x)*log(cos(2*x))|         
16*|3 - 2*log(cos(2*x)) + ----------- - -------------------------|*sin(2*x)
   |                          2                    2             |         
   \                       cos (2*x)            cos (2*x)        /         
---------------------------------------------------------------------------
                                  cos(2*x)                                 
$$\frac{16 \left(- \frac{2 \log{\left(\cos{\left(2 x \right)} \right)} \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} - 2 \log{\left(\cos{\left(2 x \right)} \right)} + \frac{3 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 3\right) \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$