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Derivative of:
Derivative of x^n
Derivative of 9
Derivative of x^2*e^(-x)
Derivative of (x+2)^2
Identical expressions
ln^ two (cos2x)
ln squared ( co sinus of e of 2x)
ln to the power of two ( co sinus of e of 2x)
ln2(cos2x)
ln2cos2x
ln²(cos2x)
ln to the power of 2(cos2x)
ln^2cos2x

The derivative of the function/ ln^2(cos2x)

Derivative of ln^2(cos2x)

The solution

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   2          
log (cos(2*x))

$$\log{\left(\cos{\left(2 x \right)} \right)}^{2}$$

log(cos(2*x))^2

Detail solution

Let $u = \log{\left(\cos{\left(2 x \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\cos{\left(2 x \right)} \right)}$ :
1. Let $u = \cos{\left(2 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 x \right)}$
  The result of the chain rule is:
  
  $- \frac{2 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$
The result of the chain rule is:

$- \frac{4 \log{\left(\cos{\left(2 x \right)} \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$
Now simplify:

$- 4 \log{\left(\cos{\left(2 x \right)} \right)} \tan{\left(2 x \right)}$

The answer is:

$- 4 \log{\left(\cos{\left(2 x \right)} \right)} \tan{\left(2 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-4*log(cos(2*x))*sin(2*x)
-------------------------
         cos(2*x)

$$- \frac{4 \log{\left(\cos{\left(2 x \right)} \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$

Simplify

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)       /

$$8 \left(- \frac{\log{\left(\cos{\left(2 x \right)} \right)} \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} - \log{\left(\cos{\left(2 x \right)} \right)} + \frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}\right)$$

Simplify

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)

$$\frac{16 \left(- \frac{2 \log{\left(\cos{\left(2 x \right)} \right)} \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} - 2 \log{\left(\cos{\left(2 x \right)} \right)} + \frac{3 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 3\right) \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$

Simplify

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Derivative of ln^2(cos2x)

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The solution