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Derivative of:
Derivative of e^x/x^2
Derivative of 5x^3
Derivative of 5*x^2-2/sqrt(x)+sin(pi/4)
Derivative of 4*sqrt(x)
Identical expressions
((x^ four *log(two *x- four ^sqrt(x)))/(sin(eight *x^ two + four)^(three)))+acos(one)/x
((x to the power of 4 multiply by logarithm of (2 multiply by x minus 4 to the power of square root of (x))) divide by ( sinus of (8 multiply by x squared plus 4) to the power of (3))) plus arc co sinus of e of ine of (1) divide by x
((x to the power of four multiply by logarithm of (two multiply by x minus four to the power of square root of (x))) divide by ( sinus of (eight multiply by x to the power of two plus four) to the power of (three))) plus arc co sinus of e of ine of (one) divide by x
((x^4*log(2*x-4^√(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x
((x4*log(2*x-4sqrt(x)))/(sin(8*x2+4)(3)))+acos(1)/x
x4*log2*x-4sqrtx/sin8*x2+43+acos1/x
((x⁴*log(2*x-4^sqrt(x)))/(sin(8*x²+4)^(3)))+acos(1)/x
((x to the power of 4*log(2*x-4 to the power of sqrt(x)))/(sin(8*x to the power of 2+4) to the power of (3)))+acos(1)/x
((x^4log(2x-4^sqrt(x)))/(sin(8x^2+4)^(3)))+acos(1)/x
((x4log(2x-4sqrt(x)))/(sin(8x2+4)(3)))+acos(1)/x
x4log2x-4sqrtx/sin8x2+43+acos1/x
x^4log2x-4^sqrtx/sin8x^2+4^3+acos1/x
((x^4*log(2*x-4^sqrt(x))) divide by (sin(8*x^2+4)^(3)))+acos(1) divide by x
Similar expressions
((x^4*log(2*x+4^sqrt(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x
((x^4*log(2*x-4^sqrt(x)))/(sin(8*x^2-4)^(3)))+acos(1)/x
((x^4*log(2*x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))-acos(1)/x
((x^4*log(2*x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))+arccos(1)/x

The derivative of the function/ ((x^4*log(2*x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x

Derivative of ((x^4log(2x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x

The solution

You have entered [src]

      /         ___\          
 4    |       \/ x |          
x *log\2*x - 4     /   acos(1)
-------------------- + -------
      3/   2    \         x   
   sin \8*x  + 4/

$$\frac{x^{4} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}}{\sin^{3}{\left(8 x^{2} + 4 \right)}} + \frac{\operatorname{acos}{\left(1 \right)}}{x}$$

(x^4*log(2*x - 4^(sqrt(x))))/sin(8*x^2 + 4)^3 + acos(1)/x

Detail solution

Differentiate $\frac{x^{4} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}}{\sin^{3}{\left(8 x^{2} + 4 \right)}} + \frac{\operatorname{acos}{\left(1 \right)}}{x}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x^{4} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}$ and $g{\left(x \right)} = \sin^{3}{\left(8 x^{2} + 4 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x^{4}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
    $g{\left(x \right)} = \log{\left(- 4^{\sqrt{x}} + 2 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = - 4^{\sqrt{x}} + 2 x$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 4^{\sqrt{x}} + 2 x\right)$ :
      1. Differentiate $- 4^{\sqrt{x}} + 2 x$ 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: $2$
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = \sqrt{x}$ .
        
        $\frac{d}{d u} 4^{u} = 4^{u} \log{\left(4 \right)}$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
        
        Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
        
        The result of the chain rule is:
        
        $\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}}$
        
        So, the result is: $- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}}$
        
        The result is: $- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}} + 2$
      The result of the chain rule is:
      
      $\frac{- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}} + 2}{- 4^{\sqrt{x}} + 2 x}$
    The result is: $\frac{x^{4} \left(- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}} + 2\right)}{- 4^{\sqrt{x}} + 2 x} + 4 x^{3} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \sin{\left(8 x^{2} + 4 \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(8 x^{2} + 4 \right)}$ :
    1. Let $u = 8 x^{2} + 4$ .
    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 $\frac{d}{d x} \left(8 x^{2} + 4\right)$ :
      1. Differentiate $8 x^{2} + 4$ term by term:
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $16 x$
        
        The derivative of the constant $4$ is zero.
        
        The result is: $16 x$
      The result of the chain rule is:
      
      $16 x \cos{\left(8 x^{2} + 4 \right)}$
    The result of the chain rule is:
    
    $48 x \sin^{2}{\left(8 x^{2} + 4 \right)} \cos{\left(8 x^{2} + 4 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- 48 x^{5} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \sin^{2}{\left(8 x^{2} + 4 \right)} \cos{\left(8 x^{2} + 4 \right)} + \left(\frac{x^{4} \left(- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}} + 2\right)}{- 4^{\sqrt{x}} + 2 x} + 4 x^{3} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \sin^{3}{\left(8 x^{2} + 4 \right)}}{\sin^{6}{\left(8 x^{2} + 4 \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  So, the result is: $- \frac{\operatorname{acos}{\left(1 \right)}}{x^{2}}$
The result is: $\frac{- 48 x^{5} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \sin^{2}{\left(8 x^{2} + 4 \right)} \cos{\left(8 x^{2} + 4 \right)} + \left(\frac{x^{4} \left(- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}} + 2\right)}{- 4^{\sqrt{x}} + 2 x} + 4 x^{3} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \sin^{3}{\left(8 x^{2} + 4 \right)}}{\sin^{6}{\left(8 x^{2} + 4 \right)}} - \frac{\operatorname{acos}{\left(1 \right)}}{x^{2}}$
Now simplify:

$\frac{- 48 x^{5} \left(4^{\sqrt{x}} - 2 x\right) \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos{\left(8 x^{2} + 4 \right)} + \frac{\left(x^{\frac{7}{2}} \left(4^{\sqrt{x}} \log{\left(4 \right)} - 4 \sqrt{x}\right) + 8 x^{3} \left(4^{\sqrt{x}} - 2 x\right) \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \sin{\left(8 x^{2} + 4 \right)}}{2}}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{4}{\left(8 x^{2} + 4 \right)}}$

The answer is:

$\frac{- 48 x^{5} \left(4^{\sqrt{x}} - 2 x\right) \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos{\left(8 x^{2} + 4 \right)} + \frac{\left(x^{\frac{7}{2}} \left(4^{\sqrt{x}} \log{\left(4 \right)} - 4 \sqrt{x}\right) + 8 x^{3} \left(4^{\sqrt{x}} - 2 x\right) \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \sin{\left(8 x^{2} + 4 \right)}}{2}}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{4}{\left(8 x^{2} + 4 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                            /       ___       \                                                  
                            |     \/ x        |                                                  
                          4 |    4     *log(4)|                                                  
                         x *|2 - -------------|                                                  
        /         ___\      |           ___   |                                                  
   3    |       \/ x |      \       2*\/ x    /                                                  
4*x *log\2*x - 4     / + ----------------------                                                  
                                       ___                                         /         ___\
                                     \/ x                       5    /   2    \    |       \/ x |
                              2*x - 4             acos(1)   48*x *cos\8*x  + 4/*log\2*x - 4     /
----------------------------------------------- - ------- - -------------------------------------
                    3/   2    \                       2                    4/   2    \           
                 sin \8*x  + 4/                      x                  sin \8*x  + 4/

$$- \frac{48 x^{5} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos{\left(8 x^{2} + 4 \right)}}{\sin^{4}{\left(8 x^{2} + 4 \right)}} + \frac{\frac{x^{4} \left(- \frac{4^{\sqrt{x}} \log{\left(4 \right)}}{2 \sqrt{x}} + 2\right)}{- 4^{\sqrt{x}} + 2 x} + 4 x^{3} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}}{\sin^{3}{\left(8 x^{2} + 4 \right)}} - \frac{\operatorname{acos}{\left(1 \right)}}{x^{2}}$$

Simplify

The second derivative [src]

                                            /                                                2                                                                 \                                                                                                                                                                                                                          
                                            |                            /        ___       \         /        ___       \                                     |                                                                                                                                                                                                                          
                                            |                            |      \/ x        |         |      \/ x        |      ___                            |                                                        /                          /        ___       \\                                                                                                                  
                                            |                          2 |     4     *log(4)|         |     4     *log(4)|    \/ x   2 /   1     log(4)\       |                                                        |                          |      \/ x        ||                                                                                                                  
                                            |                         x *|-4 + -------------|    16*x*|-4 + -------------|   4     *x *|- ---- + ------|*log(4)|                                                        |                          |     4     *log(4)||                                                                                                                  
                                            |      /     ___      \      |           ___    |         |           ___    |             |   3/2     x   |       |                                                        |                        x*|-4 + -------------||                                                                                                                  
                                          2 |      |   \/ x       |      \         \/ x     /         \         \/ x     /             \  x            /       |                                                        |     /     ___      \     |           ___    ||                                                                            /        ___       \                  
                                         x *|48*log\- 4      + 2*x/ - ------------------------ + ------------------------- + ----------------------------------|                                                      4 |     |   \/ x       |     \         \/ x     /|    /  /       2\\                                                          |      \/ x        |                  
                                            |                                           2                  ___                             ___                 |                                                  24*x *|8*log\- 4      + 2*x/ + ----------------------|*cos\4*\1 + 2*x //                                                        5 |     4     *log(4)|    /  /       2\\
                      /     ___      \      |                             /   ___      \                 \/ x                            \/ x                  |                               /     ___      \         |                                ___           |                                                   /     ___      \   24*x *|-4 + -------------|*cos\4*\1 + 2*x //
                 6    |   \/ x       |      |                             | \/ x       |                4      - 2*x                    4      - 2*x           |        4    /  /       2\\    |   \/ x       |         |                              \/ x            |                           6    2/  /       2\\    |   \/ x       |         |           ___    |                  
2*acos(1)   768*x *log\- 4      + 2*x/      \                             \4      - 2*x/                                                                       /   240*x *cos\4*\1 + 2*x //*log\- 4      + 2*x/         \                             4      - 2*x     /                     3072*x *cos \4*\1 + 2*x //*log\- 4      + 2*x/         \         \/ x     /                  
--------- + -------------------------- + ----------------------------------------------------------------------------------------------------------------------- - -------------------------------------------- - ------------------------------------------------------------------------ + ---------------------------------------------- - --------------------------------------------
     3             3/  /       2\\                                                              3/  /       2\\                                                                    4/  /       2\\                                              4/  /       2\\                                               5/  /       2\\                      /   ___      \                         
    x           sin \4*\1 + 2*x //                                                         4*sin \4*\1 + 2*x //                                                                 sin \4*\1 + 2*x //                                           sin \4*\1 + 2*x //                                            sin \4*\1 + 2*x //                      | \/ x       |    4/  /       2\\      
                                                                                                                                                                                                                                                                                                                                                   \4      - 2*x/*sin \4*\1 + 2*x //

$$\frac{768 x^{6} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}}{\sin^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{3072 x^{6} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos^{2}{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{5}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{24 x^{5} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right) \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{24 x^{4} \left(\frac{x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 8 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{240 x^{4} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{x^{2} \left(\frac{4^{\sqrt{x}} x^{2} \left(\frac{\log{\left(4 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(4 \right)}}{4^{\sqrt{x}} - 2 x} - \frac{x^{2} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)^{2}}{\left(4^{\sqrt{x}} - 2 x\right)^{2}} + \frac{16 x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 48 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right)}{4 \sin^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{2 \operatorname{acos}{\left(1 \right)}}{x^{3}}$$

Simplify

The third derivative [src]

                                                                                                        /                                                    2                            3                                                                                                                                                                                \                                                                                                                                                                                                                                        /                                                2                                                                 \                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                        |                                /        ___       \         /        ___       \          /        ___       \                                                                                                      /        ___       \                         |                                                                                                                                                                                                                                        |                            /        ___       \         /        ___       \                                     |                                                                                                                                                                                                                                                                                                                                                                                                  
                     /                          /        ___       \\                                   |                                |      \/ x        |         |      \/ x        |          |      \/ x        |      ___    /          2              \                ___                                    ___    |      \/ x        |                         |                                                                                                                                                             /                          /        ___       \\                           |                            |      \/ x        |         |      \/ x        |      ___                            |                                                                                    /                          /        ___       \\                                                                                                                                                                                                                                                              
                     |                          |      \/ x        ||                                   |                              2 |     4     *log(4)|       3 |     4     *log(4)|          |     4     *log(4)|    \/ x   3 | 3     log (4)   3*log(4)|              \/ x   2 /   1     log(4)\             \/ x   3 |     4     *log(4)| /   1     log(4)\       |                                                                                                                                                             |                          |      \/ x        ||                           |                          2 |     4     *log(4)|         |     4     *log(4)|    \/ x   2 /   1     log(4)\       |                                                                                    |                          |      \/ x        ||                                                                                                                                                                                                                                                              
                     |                          |     4     *log(4)||                                   |                          24*x *|-4 + -------------|    2*x *|-4 + -------------|    144*x*|-4 + -------------|   4     *x *|---- + ------- - --------|*log(4)   24*4     *x *|- ---- + ------|*log(4)   3*4     *x *|-4 + -------------|*|- ---- + ------|*log(4)|                                                                                                                                                             |                          |     4     *log(4)||                           |                         x *|-4 + -------------|    16*x*|-4 + -------------|   4     *x *|- ---- + ------|*log(4)|                                                                                    |                          |     4     *log(4)||                                                                                                                                                                                                                                                              
                     |                        x*|-4 + -------------||                                   |       /     ___      \         |           ___    |         |           ___    |          |           ___    |             | 5/2      3/2        2   |                       |   3/2     x   |                      |           ___    | |   3/2     x   |       |                                                                                                                                                             |                        x*|-4 + -------------||                           |      /     ___      \      |           ___    |         |           ___    |             |   3/2     x   |       |                                                                                    |                        x*|-4 + -------------||                                                                                                                               2                                                                                                                              
                     |     /     ___      \     |           ___    ||                                   |       |   \/ x       |         \         \/ x     /         \         \/ x     /          \         \/ x     /             \x        x          x    /                       \  x            /                      \         \/ x     / \  x            /       |                                                                                                                                                             |     /     ___      \     |           ___    ||                         3 |      |   \/ x       |      \         \/ x     /         \         \/ x     /             \  x            /       |    /  /       2\\             /        ___       \                                 |     /     ___      \     |           ___    ||                                                            /        ___       \                           /        ___       \                                                 /        ___       \                                                          
                   5 |     |   \/ x       |     \         \/ x     /|                                 x*|192*log\- 4      + 2*x/ - --------------------------- + -------------------------- + -------------------------- + -------------------------------------------- + ------------------------------------- - ---------------------------------------------------------|                                                                                                                                                           3 |     |   \/ x       |     \         \/ x     /|    /  /       2\\   24*x *|48*log\- 4      + 2*x/ - ------------------------ + ------------------------- + ----------------------------------|*cos\4*\1 + 2*x //             |      \/ x        |            5    2/  /       2\\ |     |   \/ x       |     \         \/ x     /|                                                            |      \/ x        |                           |      \/ x        |                                                 |      \/ x        |         ___                                              
              384*x *|8*log\- 4      + 2*x/ + ----------------------|                                   |                                              2                            3                   ___                                   ___                                         ___                                                        2                     |                                                                                                                                                       24*x *|8*log\- 4      + 2*x/ + ----------------------|*cos\4*\1 + 2*x //         |                                           2                  ___                             ___                 |                             6 |     4     *log(4)|      1536*x *cos \4*\1 + 2*x //*|8*log\- 4      + 2*x/ + ----------------------|                                                          4 |     4     *log(4)|    /  /       2\\       5 |     4     *log(4)|     /  /       2\\         6    2/  /       2\\ |     4     *log(4)|       \/ x   5 /   1     log(4)\    /  /       2\\       
                     |                                ___           |              /     ___      \     |                                /   ___      \               /   ___      \                  \/ x                                  \/ x                                        \/ x                                           /   ___      \                      |                                   /     ___      \                                  /     ___      \                               /     ___      \         |                                ___           |                           |                             /   ___      \                 \/ x                            \/ x                  |                        768*x *|-4 + -------------|                                 |                                ___           |                                  /     ___      \   240*x *|-4 + -------------|*cos\4*\1 + 2*x //   12*x *|-4 + -------------| *cos\4*\1 + 2*x //   3072*x *cos \4*\1 + 2*x //*|-4 + -------------|   12*4     *x *|- ---- + ------|*cos\4*\1 + 2*x //*log(4)
                     |                              \/ x            |         5    |   \/ x       |     |                                | \/ x       |               | \/ x       |                 4      - 2*x                          4      - 2*x                                4      - 2*x                                    | \/ x       |                      |           7    3/  /       2\\    |   \/ x       |           7    /  /       2\\    |   \/ x       |        3    /  /       2\\    |   \/ x       |         |                              \/ x            |                           |                             | \/ x       |                4      - 2*x                    4      - 2*x           |                               |           ___    |                                 |                              \/ x            |          5    2/  /       2\\    |   \/ x       |          |           ___    |                           |           ___    |                                                 |           ___    |                |   3/2     x   |                         
  6*acos(1)          \                             4      - 2*x     /   8448*x *log\- 4      + 2*x/     \                                \4      - 2*x/               \4      - 2*x/                                                                                                                                                   \4      - 2*x/                      /   245760*x *cos \4*\1 + 2*x //*log\- 4      + 2*x/   135168*x *cos\4*\1 + 2*x //*log\- 4      + 2*x/   960*x *cos\4*\1 + 2*x //*log\- 4      + 2*x/         \                             4      - 2*x     /                           \                             \4      - 2*x/                                                                       /                               \         \/ x     /                                 \                             4      - 2*x     /   33792*x *cos \4*\1 + 2*x //*log\- 4      + 2*x/          \         \/ x     /                           \         \/ x     /                                                 \         \/ x     /                \  x            /                         
- --------- + ------------------------------------------------------- + --------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------ - ----------------------------------------------- - -------------------------------------------- - ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------- + --------------------------------------------------------------------------- + ----------------------------------------------- - --------------------------------------------- + --------------------------------------------- + ----------------------------------------------- - -------------------------------------------------------
       4                            3/  /       2\\                             3/  /       2\\                                                                                                                                     3/  /       2\\                                                                                                                                              6/  /       2\\                                    4/  /       2\\                                 4/  /       2\\                                              4/  /       2\\                                                                                              4/  /       2\\                                                                /   ___      \                                                      5/  /       2\\                                                 5/  /       2\\                       /   ___      \                                                2                                  /   ___      \                                        /   ___      \                              
      x                          sin \4*\1 + 2*x //                          sin \4*\1 + 2*x //                                                                                                                                8*sin \4*\1 + 2*x //                                                                                                                                           sin \4*\1 + 2*x //                                 sin \4*\1 + 2*x //                              sin \4*\1 + 2*x //                                           sin \4*\1 + 2*x //                                                                                           sin \4*\1 + 2*x //                                                                | \/ x       |    3/  /       2\\                                sin \4*\1 + 2*x //                                              sin \4*\1 + 2*x //                       | \/ x       |    4/  /       2\\               /   ___      \                                   | \/ x       |    5/  /       2\\                     | \/ x       |    4/  /       2\\           
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             \4      - 2*x/*sin \4*\1 + 2*x //                                                                                                                                         \4      - 2*x/*sin \4*\1 + 2*x //               | \/ x       |     4/  /       2\\               \4      - 2*x/*sin \4*\1 + 2*x //                     \4      - 2*x/*sin \4*\1 + 2*x //           
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       \4      - 2*x/ *sin \4*\1 + 2*x //

$$- \frac{12 \cdot 4^{\sqrt{x}} x^{5} \left(\frac{\log{\left(4 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(4 \right)} \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{135168 x^{7} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{245760 x^{7} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{6}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{768 x^{6} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{3072 x^{6} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right) \cos^{2}{\left(4 \left(2 x^{2} + 1\right) \right)}}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{5}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{384 x^{5} \left(\frac{x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 8 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right)}{\sin^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{1536 x^{5} \left(\frac{x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 8 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \cos^{2}{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{5}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{8448 x^{5} \log{\left(- 4^{\sqrt{x}} + 2 x \right)}}{\sin^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{33792 x^{5} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos^{2}{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{5}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{12 x^{5} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)^{2} \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\left(4^{\sqrt{x}} - 2 x\right)^{2} \sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{240 x^{4} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right) \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\left(4^{\sqrt{x}} - 2 x\right) \sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{24 x^{3} \left(\frac{x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 8 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{24 x^{3} \left(\frac{4^{\sqrt{x}} x^{2} \left(\frac{\log{\left(4 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(4 \right)}}{4^{\sqrt{x}} - 2 x} - \frac{x^{2} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)^{2}}{\left(4^{\sqrt{x}} - 2 x\right)^{2}} + \frac{16 x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 48 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right) \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{960 x^{3} \log{\left(- 4^{\sqrt{x}} + 2 x \right)} \cos{\left(4 \left(2 x^{2} + 1\right) \right)}}{\sin^{4}{\left(4 \left(2 x^{2} + 1\right) \right)}} + \frac{x \left(\frac{4^{\sqrt{x}} x^{3} \left(- \frac{3 \log{\left(4 \right)}}{x^{2}} + \frac{\log{\left(4 \right)}^{2}}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right) \log{\left(4 \right)}}{4^{\sqrt{x}} - 2 x} - \frac{3 \cdot 4^{\sqrt{x}} x^{3} \left(\frac{\log{\left(4 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right) \log{\left(4 \right)}}{\left(4^{\sqrt{x}} - 2 x\right)^{2}} + \frac{24 \cdot 4^{\sqrt{x}} x^{2} \left(\frac{\log{\left(4 \right)}}{x} - \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(4 \right)}}{4^{\sqrt{x}} - 2 x} + \frac{2 x^{3} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)^{3}}{\left(4^{\sqrt{x}} - 2 x\right)^{3}} - \frac{24 x^{2} \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)^{2}}{\left(4^{\sqrt{x}} - 2 x\right)^{2}} + \frac{144 x \left(\frac{4^{\sqrt{x}} \log{\left(4 \right)}}{\sqrt{x}} - 4\right)}{4^{\sqrt{x}} - 2 x} + 192 \log{\left(- 4^{\sqrt{x}} + 2 x \right)}\right)}{8 \sin^{3}{\left(4 \left(2 x^{2} + 1\right) \right)}} - \frac{6 \operatorname{acos}{\left(1 \right)}}{x^{4}}$$

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Derivative of ((x^4log(2x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x

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Derivative of ((x^4*log(2*x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x

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Derivative of ((x^4log(2x-4^sqrt(x)))/(sin(8*x^2+4)^(3)))+acos(1)/x