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Derivative of:
Derivative of sin(3*x)^(2)
Derivative of log(x)*cos(x)
Derivative of log(x^2*e^x)
Derivative of lnx-x
Identical expressions
(log(x*cos(x))*(e^(three *x+cos(two *x))+x^ two)^ one)/(sin(x/ two -e^(two *x)))
( logarithm of (x multiply by co sinus of e of (x)) multiply by (e to the power of (3 multiply by x plus co sinus of e of (2 multiply by x)) plus x squared ) to the power of 1) divide by ( sinus of (x divide by 2 minus e to the power of (2 multiply by x)))
( logarithm of (x multiply by co sinus of e of (x)) multiply by (e to the power of (three multiply by x plus co sinus of e of (two multiply by x)) plus x to the power of two) to the power of one) divide by ( sinus of (x divide by two minus e to the power of (two multiply by x)))
(log(x*cos(x))*(e(3*x+cos(2*x))+x2)1)/(sin(x/2-e(2*x)))
logx*cosx*e3*x+cos2*x+x21/sinx/2-e2*x
(log(x*cos(x))*(e^(3*x+cos(2*x))+x²)^1)/(sin(x/2-e^(2*x)))
(log(x*cos(x))*(e to the power of (3*x+cos(2*x))+x to the power of 2) to the power of 1)/(sin(x/2-e to the power of (2*x)))
(log(xcos(x))(e^(3x+cos(2x))+x^2)^1)/(sin(x/2-e^(2x)))
(log(xcos(x))(e(3x+cos(2x))+x2)1)/(sin(x/2-e(2x)))
logxcosxe3x+cos2x+x21/sinx/2-e2x
logxcosxe^3x+cos2x+x^2^1/sinx/2-e^2x
(log(x*cos(x))*(e^(3*x+cos(2*x))+x^2)^1) divide by (sin(x divide by 2-e^(2*x)))
Similar expressions
(log(x*cos(x))*(e^(3*x+cos(2*x))-x^2)^1)/(sin(x/2-e^(2*x)))
(log(x*cos(x))*(e^(3*x+cos(2*x))+x^2)^1)/(sin(x/2+e^(2*x)))
(log(x*cos(x))*(e^(3*x-cos(2*x))+x^2)^1)/(sin(x/2-e^(2*x)))
(log(x*cosx)*(e^(3*x+cos(2*x))+x^2)^1)/(sin(x/2-e^(2*x)))

The derivative of the function/ (log(x*cos(x))*(e^(3*x+cos(2*x))+x^2)^1)/(sin(x/2-e^(2*x)))

Derivative of (log(xcos(x))(e^(3x+cos(2x))+x^2)^1)/(sin(x/2-e^(2*x)))

The solution

You have entered [src]

                                    1
              / 3*x + cos(2*x)    2\ 
log(x*cos(x))*\E               + x / 
-------------------------------------
               /x    2*x\            
            sin|- - E   |            
               \2       /

$$\frac{\left(e^{3 x + \cos{\left(2 x \right)}} + x^{2}\right)^{1} \log{\left(x \cos{\left(x \right)} \right)}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}$$

(log(x*cos(x))*(E^(3*x + cos(2*x)) + x^2)^1)/sin(x/2 - E^(2*x))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{x}{2} - e^{2 x} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{2} + e^{3 x + \cos{\left(2 x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $x^{2} + e^{3 x + \cos{\left(2 x \right)}}$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. Let $u = 3 x + \cos{\left(2 x \right)}$ .
    3. The derivative of $e^{u}$ is itself.
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + \cos{\left(2 x \right)}\right)$ :
      1. Differentiate $3 x + \cos{\left(2 x \right)}$ term by term:
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
        
        Let $u = 2 x$ .
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        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 is: $3 - 2 \sin{\left(2 x \right)}$
      The result of the chain rule is:
      
      $\left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}$
    The result is: $2 x + \left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}$
  $g{\left(x \right)} = \log{\left(x \cos{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x \cos{\left(x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x \cos{\left(x \right)}$ :
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      The result is: $- x \sin{\left(x \right)} + \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{- x \sin{\left(x \right)} + \cos{\left(x \right)}}{x \cos{\left(x \right)}}$
  The result is: $\left(2 x + \left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} + \frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{x}{2} - e^{2 x}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\frac{x}{2} - e^{2 x}\right)$ :
  1. Differentiate $\frac{x}{2} - e^{2 x}$ term by term:
    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: $\frac{1}{2}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = 2 x$ .
      2. The derivative of $e^{u}$ is itself.
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
        
        The result of the chain rule is:
        
        $2 e^{2 x}$
      So, the result is: $- 2 e^{2 x}$
    The result is: $\frac{1}{2} - 2 e^{2 x}$
  The result of the chain rule is:
  
  $\left(\frac{1}{2} - 2 e^{2 x}\right) \cos{\left(\frac{x}{2} - e^{2 x} \right)}$
Now plug in to the quotient rule:

$\frac{- \left(\frac{1}{2} - 2 e^{2 x}\right) \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(\frac{x}{2} - e^{2 x} \right)} + \left(\left(2 x + \left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} + \frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}}\right) \sin{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)}}$
Now simplify:

$\frac{\frac{x \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(4 e^{2 x} - 1\right) \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{2} + \left(x \left(2 x + \left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right) \sin{\left(\frac{x}{2} - e^{2 x} \right)}}{x \sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)} \cos{\left(x \right)}}$

The answer is:

$\frac{\frac{x \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(4 e^{2 x} - 1\right) \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{2} + \left(x \left(2 x + \left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)\right) \sin{\left(\frac{x}{2} - e^{2 x} \right)}}{x \sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)} \cos{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                                         / 3*x + cos(2*x)    2\                                                                                       
/                        3*x + cos(2*x)\                 \E               + x /*(-x*sin(x) + cos(x))   /1      2*x\ / 3*x + cos(2*x)    2\    /x    2*x\              
\2*x + (3 - 2*sin(2*x))*e              /*log(x*cos(x)) + -------------------------------------------   |- - 2*e   |*\E               + x /*cos|- - E   |*log(x*cos(x))
                                                                           x*cos(x)                    \2         /                           \2       /              
---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------
                                              /x    2*x\                                                                           2/x    2*x\                        
                                           sin|- - E   |                                                                        sin |- - E   |                        
                                              \2       /                                                                            \2       /

$$- \frac{\left(\frac{1}{2} - 2 e^{2 x}\right) \left(e^{3 x + \cos{\left(2 x \right)}} + x^{2}\right) \log{\left(x \cos{\left(x \right)} \right)} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)}} + \frac{\left(2 x + \left(3 - 2 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} + \frac{\left(e^{3 x + \cos{\left(2 x \right)}} + x^{2}\right) \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}$$

Simplify

The second derivative [src]

                                                                                                             /                                2                                       \                                                                                                                                                                                                                                                                                                                         
                                                                                                             |                   /        2*x\     2/x    2*x\         /x    2*x\  2*x|                                                                                                                                                                                                                                                                                                                         
                                                                                                             |             2   2*\-1 + 4*e   / *cos |- - e   |   16*cos|- - e   |*e   |                                                                                                                                                                                                                                                                                                                         
                                                                                      / 2    3*x + cos(2*x)\ |/        2*x\                         \2       /         \2       /     |                                                                                                                                                                                                                                                                                                                         
                                                                                      \x  + e              /*|\-1 + 4*e   /  + ------------------------------- + ---------------------|*log(x*cos(x))                 /                                                          / 2    3*x + cos(2*x)\                     \                                                                                                                                                                                   
                                                                                                             |                             2/x    2*x\                  /x    2*x\    |                 /        2*x\ |/                         3*x + cos(2*x)\                 \x  + e              /*(-cos(x) + x*sin(x))|    /x    2*x\   / 2    3*x + cos(2*x)\ /                      -cos(x) + x*sin(x)   (-cos(x) + x*sin(x))*sin(x)\                                                                   
                                                                                                             |                          sin |- - e   |               sin|- - e   |    |                 \-1 + 4*e   /*|\2*x - (-3 + 2*sin(2*x))*e              /*log(x*cos(x)) - -------------------------------------------|*cos|- - e   |   \x  + e              /*|2*sin(x) + x*cos(x) - ------------------ + ---------------------------|                          /                         3*x + cos(2*x)\
/                     2  3*x + cos(2*x)               3*x + cos(2*x)\                                        \                              \2       /                  \2       /    /                               \                                                                            x*cos(x)                 /    \2       /                          \                              x                       cos(x)          /   2*(-cos(x) + x*sin(x))*\2*x - (-3 + 2*sin(2*x))*e              /
\2 + (-3 + 2*sin(2*x)) *e               - 4*cos(2*x)*e              /*log(x*cos(x)) + --------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------
                                                                                                                                             4                                                                                                                        /x    2*x\                                                                                                          x*cos(x)                                                                          x*cos(x)                            
                                                                                                                                                                                                                                                                   sin|- - e   |                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                      \2       /                                                                                                                                                                                                                                
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                    /x    2*x\                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                 sin|- - e   |                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                    \2       /

$$\frac{\frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(\left(4 e^{2 x} - 1\right)^{2} + \frac{2 \left(4 e^{2 x} - 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)}} + \frac{16 e^{2 x} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)}}{4} + \frac{\left(\left(2 x - \left(2 \sin{\left(2 x \right)} - 3\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} - \frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}}\right) \left(4 e^{2 x} - 1\right) \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}} + \left(\left(2 \sin{\left(2 x \right)} - 3\right)^{2} e^{3 x + \cos{\left(2 x \right)}} - 4 e^{3 x + \cos{\left(2 x \right)}} \cos{\left(2 x \right)} + 2\right) \log{\left(x \cos{\left(x \right)} \right)} - \frac{2 \left(2 x - \left(2 \sin{\left(2 x \right)} - 3\right) e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}} - \frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)}{x \cos{\left(x \right)}}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}$$

Simplify

The third derivative [src]

                                                                                                          /                                2                                       \                                                                                                                              /                                       3                                3                                                                              \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                          |                   /        2*x\     2/x    2*x\         /x    2*x\  2*x|                                                                                                                              |                          /        2*x\     /x    2*x\     /        2*x\     3/x    2*x\         /x    2*x\  2*x         2/x    2*x\ /        2*x\  2*x|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
  /                                                          / 2    3*x + cos(2*x)\                     \ |             2   2*\-1 + 4*e   / *cos |- - e   |   16*cos|- - e   |*e   |                                                                                                                              |                        5*\-1 + 4*e   / *cos|- - e   |   6*\-1 + 4*e   / *cos |- - e   |   64*cos|- - e   |*e      96*cos |- - e   |*\-1 + 4*e   /*e   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
  |/                         3*x + cos(2*x)\                 \x  + e              /*(-cos(x) + x*sin(x))| |/        2*x\                         \2       /         \2       /     |                                                                                                       / 2    3*x + cos(2*x)\ |   /        2*x\  2*x                       \2       /                        \2       /         \2       /               \2       /                   |                                                                                                                                                                                                                                                                /                                                                             2                                                               \                   /                                                                                        / 2    3*x + cos(2*x)\ /                      -cos(x) + x*sin(x)   (-cos(x) + x*sin(x))*sin(x)\                                                                   \              
3*|\2*x - (-3 + 2*sin(2*x))*e              /*log(x*cos(x)) - -------------------------------------------|*|\-1 + 4*e   /  + ------------------------------- + ---------------------|                                                                                                       \x  + e              /*|48*\-1 + 4*e   /*e    + ------------------------------ + ------------------------------- + --------------------- + ------------------------------------|*log(x*cos(x))                                                                                                                                                                                                                           / 2    3*x + cos(2*x)\ |-cos(x) + x*sin(x)   2*sin(x) + x*cos(x)   (2*sin(x) + x*cos(x))*sin(x)   sin (x)*(-cos(x) + x*sin(x))   (-cos(x) + x*sin(x))*sin(x)         |                   |                                                                                        \x  + e              /*|2*sin(x) + x*cos(x) - ------------------ + ---------------------------|                          /                         3*x + cos(2*x)\|              
  \                                                                            x*cos(x)                 / |                             2/x    2*x\                  /x    2*x\    |                                                                                                                              |                                   /x    2*x\                        3/x    2*x\                  /x    2*x\                     2/x    2*x\           |                                                                                                                  /                         3*x + cos(2*x)\ /                      -cos(x) + x*sin(x)   (-cos(x) + x*sin(x))*sin(x)\   2*\x  + e              /*|------------------ - ------------------- + ---------------------------- + ---------------------------- - --------------------------- + cos(x)|     /        2*x\ |  /                     2  3*x + cos(2*x)               3*x + cos(2*x)\                                        \                              x                       cos(x)          /   2*(-cos(x) + x*sin(x))*\2*x - (-3 + 2*sin(2*x))*e              /|    /x    2*x\
                                                                                                          |                          sin |- - e   |               sin|- - e   |    |                                                                                                                              |                                sin|- - e   |                     sin |- - e   |               sin|- - e   |                  sin |- - e   |           |                                        /                     2  3*x + cos(2*x)               3*x + cos(2*x)\   3*\2*x - (-3 + 2*sin(2*x))*e              /*|2*sin(x) + x*cos(x) - ------------------ + ---------------------------|                            |         2                    x                       cos(x)                           2                           x*cos(x)                  |   3*\-1 + 4*e   /*|- \2 + (-3 + 2*sin(2*x)) *e               - 4*cos(2*x)*e              /*log(x*cos(x)) + ----------------------------------------------------------------------------------------------- + ----------------------------------------------------------------|*cos|- - e   |
                                                                                                          \                              \2       /                  \2       /    /   /                   3                                             \  3*x + cos(2*x)                                        \                                   \2       /                         \2       /                  \2       /                      \2       /           /                 3*(-cos(x) + x*sin(x))*\2 + (-3 + 2*sin(2*x)) *e               - 4*cos(2*x)*e              /                                               \                              x                       cos(x)          /                            \        x                                                                           cos (x)                                                  /                   \                                                                                                                                    x*cos(x)                                                                          x*cos(x)                            /    \2       /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + \- (-3 + 2*sin(2*x))  + 8*sin(2*x) + 12*(-3 + 2*sin(2*x))*cos(2*x)/*e              *log(x*cos(x)) + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                         4                                                                                                                                                                                                                                                                                               8                                                                                                                                            x*cos(x)                                                                                                   x*cos(x)                                                                                                                                         x*cos(x)                                                                                                                                                                                                                             /x    2*x\                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2*sin|- - e   |                                                                                                                                      
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        /x    2*x\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
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$$\frac{\frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(\frac{5 \left(4 e^{2 x} - 1\right)^{3} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}} + \frac{6 \left(4 e^{2 x} - 1\right)^{3} \cos^{3}{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin^{3}{\left(\frac{x}{2} - e^{2 x} \right)}} + 48 \left(4 e^{2 x} - 1\right) e^{2 x} + \frac{96 \left(4 e^{2 x} - 1\right) e^{2 x} \cos^{2}{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)}} + \frac{64 e^{2 x} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)}}{8} + \frac{3 \left(\left(2 x - \left(2 \sin{\left(2 x \right)} - 3\right) e^{3 x + \cos{\left(2 x \right)}}\right) \log{\left(x \cos{\left(x \right)} \right)} - \frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}}\right) \left(\left(4 e^{2 x} - 1\right)^{2} + \frac{2 \left(4 e^{2 x} - 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin^{2}{\left(\frac{x}{2} - e^{2 x} \right)}} + \frac{16 e^{2 x} \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}\right)}{4} - \frac{3 \left(4 e^{2 x} - 1\right) \left(- \left(\left(2 \sin{\left(2 x \right)} - 3\right)^{2} e^{3 x + \cos{\left(2 x \right)}} - 4 e^{3 x + \cos{\left(2 x \right)}} \cos{\left(2 x \right)} + 2\right) \log{\left(x \cos{\left(x \right)} \right)} + \frac{2 \left(2 x - \left(2 \sin{\left(2 x \right)} - 3\right) e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}} + \frac{\left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)}{x \cos{\left(x \right)}}\right) \cos{\left(\frac{x}{2} - e^{2 x} \right)}}{2 \sin{\left(\frac{x}{2} - e^{2 x} \right)}} + \left(- \left(2 \sin{\left(2 x \right)} - 3\right)^{3} + 12 \left(2 \sin{\left(2 x \right)} - 3\right) \cos{\left(2 x \right)} + 8 \sin{\left(2 x \right)}\right) e^{3 x + \cos{\left(2 x \right)}} \log{\left(x \cos{\left(x \right)} \right)} - \frac{3 \left(2 x - \left(2 \sin{\left(2 x \right)} - 3\right) e^{3 x + \cos{\left(2 x \right)}}\right) \left(x \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2 \sin{\left(x \right)} - \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)}{x \cos{\left(x \right)}} - \frac{2 \left(x^{2} + e^{3 x + \cos{\left(2 x \right)}}\right) \left(\frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \cos{\left(x \right)} - \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x} + \frac{x \sin{\left(x \right)} - \cos{\left(x \right)}}{x^{2}}\right)}{x \cos{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\left(2 \sin{\left(2 x \right)} - 3\right)^{2} e^{3 x + \cos{\left(2 x \right)}} - 4 e^{3 x + \cos{\left(2 x \right)}} \cos{\left(2 x \right)} + 2\right)}{x \cos{\left(x \right)}}}{\sin{\left(\frac{x}{2} - e^{2 x} \right)}}$$

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Derivative of (log(xcos(x))(e^(3x+cos(2x))+x^2)^1)/(sin(x/2-e^(2*x)))

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Derivative of (log(xcos(x))(e^(3x+cos(2x))+x^2)^1)/(sin(x/2-e^(2*x)))