Derivative of ln*sqrt(cos2x)

The solution

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

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

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \sqrt{\cos{\left(2 x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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 is: $- \frac{\log{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + \frac{\sqrt{\cos{\left(2 x \right)}}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

 /  __________   /                     2      \                        \
 |\/ cos(2*x)    |    __________    sin (2*x) |            2*sin(2*x)  |
-|------------ + |2*\/ cos(2*x)  + -----------|*log(x) + --------------|
 |      2        |                    3/2     |              __________|
 \     x         \                 cos   (2*x)/          x*\/ cos(2*x) /

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

Simplify

The third derivative [src]

    /                     2      \                                      /         2     \                
    |    __________    sin (2*x) |                                      |    3*sin (2*x)|                
  3*|2*\/ cos(2*x)  + -----------|                                      |2 + -----------|*log(x)*sin(2*x)
    |                    3/2     |       __________                     |        2      |                
    \                 cos   (2*x)/   2*\/ cos(2*x)       3*sin(2*x)     \     cos (2*x) /                
- -------------------------------- + -------------- + --------------- - ---------------------------------
                 x                          3          2   __________                __________          
                                           x          x *\/ cos(2*x)               \/ cos(2*x)

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

Simplify

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Derivative of ln*sqrt(cos2x)

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The solution