Derivative of lnsqrt(cos(2x))

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   /  __________\
log\\/ cos(2*x) /

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

log(sqrt(cos(2*x)))

Detail solution

Let $u = \sqrt{\cos{\left(2 x \right)}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\cos{\left(2 x \right)}}$ :
1. Let $u = \cos{\left(2 x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
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:
  
  $- \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   /       2     \
   |    sin (2*x)|
-2*|1 + ---------|
   |       2     |
   \    cos (2*x)/

$$- 2 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right)$$

Simplify

The third derivative [src]

   /       2     \         
   |    sin (2*x)|         
-8*|1 + ---------|*sin(2*x)
   |       2     |         
   \    cos (2*x)/         
---------------------------
          cos(2*x)

$$- \frac{8 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$

Simplify

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Derivative of lnsqrt(cos(2x))

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The solution