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Derivative of:
Derivative of (x-1)/(x+1)
Derivative of 4x^2
Derivative of sqrt(x^3)
Derivative of sqrt(x^2)
Identical expressions
(x^ two *sqrt(x+ one))/((x- one)^ three *(one / four ^sqrt(five *x- one)))
(x squared multiply by square root of (x plus 1)) divide by ((x minus 1) cubed multiply by (1 divide by 4 to the power of square root of (5 multiply by x minus 1)))
(x to the power of two multiply by square root of (x plus one)) divide by ((x minus one) to the power of three multiply by (one divide by four to the power of square root of (five multiply by x minus one)))
(x^2*√(x+1))/((x-1)^3*(1/4^√(5*x-1)))
(x2*sqrt(x+1))/((x-1)3*(1/4sqrt(5*x-1)))
x2*sqrtx+1/x-13*1/4sqrt5*x-1
(x²*sqrt(x+1))/((x-1)³*(1/4^sqrt(5*x-1)))
(x to the power of 2*sqrt(x+1))/((x-1) to the power of 3*(1/4 to the power of sqrt(5*x-1)))
(x^2sqrt(x+1))/((x-1)^3(1/4^sqrt(5x-1)))
(x2sqrt(x+1))/((x-1)3(1/4sqrt(5x-1)))
x2sqrtx+1/x-131/4sqrt5x-1
x^2sqrtx+1/x-1^31/4^sqrt5x-1
(x^2*sqrt(x+1)) divide by ((x-1)^3*(1 divide by 4^sqrt(5*x-1)))
Similar expressions
(x^2*sqrt(x+1))/((x+1)^3*(1/4^sqrt(5*x-1)))
(x^2*sqrt(x-1))/((x-1)^3*(1/4^sqrt(5*x-1)))
(x^2*sqrt(x+1))/((x-1)^3*(1/4^sqrt(5*x+1)))

The derivative of the function/ (x^2*sqrt(x+1))/((x-1)^3*(1/4^sqrt(5*x-1)))

Derivative of (x^2sqrt(x+1))/((x-1)^3(1/4^sqrt(5*x-1)))

The solution

You have entered [src]

      2   _______     
     x *\/ x + 1      
----------------------
             _________
       3  -\/ 5*x - 1 
(x - 1) *4

$$\frac{x^{2} \sqrt{x + 1}}{\left(\frac{1}{4}\right)^{\sqrt{5 x - 1}} \left(x - 1\right)^{3}}$$

(x^2*sqrt(x + 1))/(((x - 1)^3*(1/4)^(sqrt(5*x - 1))))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 4^{\sqrt{5 x - 1}} x^{2} \sqrt{x + 1}$ and $g{\left(x \right)} = \left(x - 1\right)^{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 4^{\sqrt{5 x - 1}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \sqrt{5 x - 1}$ .
  2. $\frac{d}{d u} 4^{u} = 4^{u} \log{\left(4 \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{5 x - 1}$ :
    1. Let $u = 5 x - 1$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 1\right)$ :
      1. Differentiate $5 x - 1$ 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: $5$
        
        The derivative of the constant $-1$ is zero.
        
        The result is: $5$
      The result of the chain rule is:
      
      $\frac{5}{2 \sqrt{5 x - 1}}$
    The result of the chain rule is:
    
    $\frac{5 \cdot 4^{\sqrt{5 x - 1}} \log{\left(4 \right)}}{2 \sqrt{5 x - 1}}$
  $g{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  $h{\left(x \right)} = \sqrt{x + 1}$ ; to find $\frac{d}{d x} h{\left(x \right)}$ :
  1. Let $u = x + 1$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
    1. Differentiate $x + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{2 \sqrt{x + 1}}$
  The result is: $\frac{5 \cdot 4^{\sqrt{5 x - 1}} x^{2} \sqrt{x + 1} \log{\left(4 \right)}}{2 \sqrt{5 x - 1}} + \frac{4^{\sqrt{5 x - 1}} x^{2}}{2 \sqrt{x + 1}} + 2 \cdot 4^{\sqrt{5 x - 1}} x \sqrt{x + 1}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 1$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $3 \left(x - 1\right)^{2}$
Now plug in to the quotient rule:

$\frac{- 3 \cdot 4^{\sqrt{5 x - 1}} x^{2} \left(x - 1\right)^{2} \sqrt{x + 1} + \left(x - 1\right)^{3} \left(\frac{5 \cdot 4^{\sqrt{5 x - 1}} x^{2} \sqrt{x + 1} \log{\left(4 \right)}}{2 \sqrt{5 x - 1}} + \frac{4^{\sqrt{5 x - 1}} x^{2}}{2 \sqrt{x + 1}} + 2 \cdot 4^{\sqrt{5 x - 1}} x \sqrt{x + 1}\right)}{\left(x - 1\right)^{6}}$
Now simplify:

$\frac{x \left(- 6 \cdot 2^{2 \sqrt{5 x - 1}} x \left(x + 1\right) \sqrt{5 x - 1} + \left(x - 1\right) \left(4 \cdot 2^{2 \sqrt{5 x - 1}} \left(x + 1\right) \sqrt{5 x - 1} + 10 \cdot 4^{\sqrt{5 x - 1}} x \left(x + 1\right) \log{\left(2 \right)} + 4^{\sqrt{5 x - 1}} x \sqrt{5 x - 1}\right)\right)}{2 \left(x - 1\right)^{4} \sqrt{x + 1} \sqrt{5 x - 1}}$

The answer is:

$\frac{x \left(- 6 \cdot 2^{2 \sqrt{5 x - 1}} x \left(x + 1\right) \sqrt{5 x - 1} + \left(x - 1\right) \left(4 \cdot 2^{2 \sqrt{5 x - 1}} \left(x + 1\right) \sqrt{5 x - 1} + 10 \cdot 4^{\sqrt{5 x - 1}} x \left(x + 1\right) \log{\left(2 \right)} + 4^{\sqrt{5 x - 1}} x \sqrt{5 x - 1}\right)\right)}{2 \left(x - 1\right)^{4} \sqrt{x + 1} \sqrt{5 x - 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                                                         /                                   _________                \
                                                  _________              |        _________               -\/ 5*x - 1         3       |
                                              2*\/ 5*x - 1   2   _______ |     -\/ 5*x - 1         2   5*4            *(x - 1) *log(4)|
   _________                                 4             *x *\/ x + 1 *|- 3*4            *(x - 1)  + -------------------------------|
 \/ 5*x - 1  /      2                    \                               |                                          _________         |
4            |     x              _______|                               \                                      2*\/ 5*x - 1          /
------------*|----------- + 2*x*\/ x + 1 | + ------------------------------------------------------------------------------------------
         3   |    _______                |                                                   6                                         
  (x - 1)    \2*\/ x + 1                 /                                            (x - 1)

$$\frac{4^{2 \sqrt{5 x - 1}} x^{2} \sqrt{x + 1} \left(\frac{5 \cdot 4^{- \sqrt{5 x - 1}} \left(x - 1\right)^{3} \log{\left(4 \right)}}{2 \sqrt{5 x - 1}} - 3 \cdot 4^{- \sqrt{5 x - 1}} \left(x - 1\right)^{2}\right)}{\left(x - 1\right)^{6}} + \frac{4^{\sqrt{5 x - 1}}}{\left(x - 1\right)^{3}} \left(\frac{x^{2}}{2 \sqrt{x + 1}} + 2 x \sqrt{x + 1}\right)$$

Simplify

The second derivative [src]

              /                                                                                                             /                                                                                          2    2                 2                                                                        \\
              |                                                                                                             |                                                          60*(-1 + x)*log(4)   25*(-1 + x) *log (4)   25*(-1 + x) *log(4)     /     5*(-1 + x)*log(4)\     /     5*(-1 + x)*log(4)\       ||
              |                                                                                                             |                                                     24 - ------------------ + -------------------- + -------------------   6*|-6 + -----------------|   5*|-6 + -----------------|*log(4)||
              |                                                                                                             |                                                               __________            -1 + 5*x                      3/2        |          __________  |     |          __________  |       ||
              |                                           /     5*(-1 + x)*log(4)\ /    _______       x    \    2   _______ |/     5*(-1 + x)*log(4)\ /    6        5*log(4)  \           \/ -1 + 5*x                                 (-1 + 5*x)           \        \/ -1 + 5*x   /     \        \/ -1 + 5*x   /       ||
              |                                         x*|-6 + -----------------|*|4*\/ 1 + x  + ---------|   x *\/ 1 + x *||-6 + -----------------|*|- ------ + ------------| - -------------------------------------------------------------------- - -------------------------- + ---------------------------------||
   __________ |                                2          |          __________  | |                _______|                ||          __________  | |  -1 + x     __________|                                  -1 + x                                            -1 + x                          __________          ||
 \/ -1 + 5*x  |    _______      2*x           x           \        \/ -1 + 5*x   / \              \/ 1 + x /                \\        \/ -1 + 5*x   / \           \/ -1 + 5*x /                                                                                                                  \/ -1 + 5*x           /|
4            *|2*\/ 1 + x  + --------- - ------------ + ---------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
              |                _______            3/2                        2*(-1 + x)                                                                                                                        4*(-1 + x)                                                                                               |
              \              \/ 1 + x    4*(1 + x)                                                                                                                                                                                                                                                                      /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                3                                                                                                                                                        
                                                                                                                                                        (-1 + x)

$$\frac{4^{\sqrt{5 x - 1}} \left(- \frac{x^{2}}{4 \left(x + 1\right)^{\frac{3}{2}}} + \frac{x^{2} \sqrt{x + 1} \left(\left(\frac{5 \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6}{x - 1}\right) \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) + \frac{5 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right)}{x - 1} - \frac{\frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}^{2}}{5 x - 1} + \frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{60 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} + 24}{x - 1}\right)}{4 \left(x - 1\right)} + \frac{2 x}{\sqrt{x + 1}} + \frac{x \left(\frac{x}{\sqrt{x + 1}} + 4 \sqrt{x + 1}\right) \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right)}{2 \left(x - 1\right)} + 2 \sqrt{x + 1}\right)}{\left(x - 1\right)^{3}}$$

Simplify

The third derivative [src]

              /                                                                                                            /                  2    2                  2                      3    3                                        3    2                  3                                                                                                         /                                     2    2                 2       \                                                           /                                     2    2                 2       \                                                                                                                                                                              /                                     2    2                 2       \                                                                     \                                 /                                                                                          2    2                 2                                                                        \\
              |                                                                                                            |      450*(-1 + x) *log (4)   450*(-1 + x) *log(4)   125*(-1 + x) *log (4)   360*(-1 + x)*log(4)   375*(-1 + x) *log (4)   375*(-1 + x) *log(4)                                                                                                  |     60*(-1 + x)*log(4)   25*(-1 + x) *log (4)   25*(-1 + x) *log(4)|      /     5*(-1 + x)*log(4)\   /    6        5*log(4)  \ |     60*(-1 + x)*log(4)   25*(-1 + x) *log (4)   25*(-1 + x) *log(4)|      /     5*(-1 + x)*log(4)\            /     5*(-1 + x)*log(4)\ /    6        5*log(4)  \         2    /     5*(-1 + x)*log(4)\       /     5*(-1 + x)*log(4)\             |     60*(-1 + x)*log(4)   25*(-1 + x) *log (4)   25*(-1 + x) *log(4)|            /     5*(-1 + x)*log(4)\ /    6        5*log(4)  \       |                                 |                                                          60*(-1 + x)*log(4)   25*(-1 + x) *log (4)   25*(-1 + x) *log(4)     /     5*(-1 + x)*log(4)\     /     5*(-1 + x)*log(4)\       ||
              |                                                                                                            |-48 - --------------------- - -------------------- + --------------------- + ------------------- + --------------------- + --------------------                                                                                               18*|24 - ------------------ + -------------------- + -------------------|   84*|-6 + -----------------|   |- ------ + ------------|*|24 - ------------------ + -------------------- + -------------------|   25*|-6 + -----------------|*log(4)   6*|-6 + -----------------|*|- ------ + ------------|   50*log (4)*|-6 + -----------------|   120*|-6 + -----------------|*log(4)   15*|24 - ------------------ + -------------------- + -------------------|*log(4)   5*|-6 + -----------------|*|- ------ + ------------|*log(4)|                                 |                                                     24 - ------------------ + -------------------- + -------------------   6*|-6 + -----------------|   5*|-6 + -----------------|*log(4)||
              |  /        2           \                              /                   2                \                |             -1 + 5*x                      3/2                     3/2             __________                     2                     5/2                                /                                  2                           \      |          __________            -1 + 5*x                      3/2   |      |          __________  |   |  -1 + x     __________| |          __________            -1 + 5*x                      3/2   |      |          __________  |            |          __________  | |  -1 + x     __________|              |          __________  |       |          __________  |             |          __________            -1 + 5*x                      3/2   |            |          __________  | |  -1 + x     __________|       |                                 |                                                               __________            -1 + 5*x                      3/2        |          __________  |     |          __________  |       ||
              |  |       x        4*x |     /     5*(-1 + x)*log(4)\ |    _______       x           8*x   |    2   _______ |                                 (-1 + 5*x)              (-1 + 5*x)              \/ -1 + 5*x            (-1 + 5*x)            (-1 + 5*x)          /     5*(-1 + x)*log(4)\ |    48        25*log(4)     25*log (4)         60*log(4)      |      \        \/ -1 + 5*x                                 (-1 + 5*x)      /      \        \/ -1 + 5*x   /   \           \/ -1 + 5*x / \        \/ -1 + 5*x                                 (-1 + 5*x)      /      \        \/ -1 + 5*x   /            \        \/ -1 + 5*x   / \           \/ -1 + 5*x /              \        \/ -1 + 5*x   /       \        \/ -1 + 5*x   /             \        \/ -1 + 5*x                                 (-1 + 5*x)      /            \        \/ -1 + 5*x   / \           \/ -1 + 5*x /       |       /    _______       x    \ |/     5*(-1 + x)*log(4)\ /    6        5*log(4)  \           \/ -1 + 5*x                                 (-1 + 5*x)           \        \/ -1 + 5*x   /     \        \/ -1 + 5*x   /       ||
              |3*|8 + -------- - -----|   3*|-6 + -----------------|*|8*\/ 1 + x  - ---------- + ---------|   x *\/ 1 + x *|----------------------------------------------------------------------------------------------------------------------------------------------- + |-6 + -----------------|*|--------- - ------------- + ---------- - ---------------------| + ------------------------------------------------------------------------- + --------------------------- - ------------------------------------------------------------------------------------------------ - ---------------------------------- - ---------------------------------------------------- + ----------------------------------- - ----------------------------------- - -------------------------------------------------------------------------------- + -----------------------------------------------------------|   3*x*|4*\/ 1 + x  + ---------|*||-6 + -----------------|*|- ------ + ------------| - -------------------------------------------------------------------- - -------------------------- + ---------------------------------||
   __________ |  |           2   1 + x|     |          __________  | |                     3/2     _______|                |                                                                           2                                                                      |          __________  | |        2             3/2    -1 + 5*x               __________|                                           2                                                    2                                                         -1 + x                                                                    3/2                                     -1 + x                                        -1 + 5*x                                  __________                                                  __________                                                           __________                       |       |                _______| ||          __________  | |  -1 + x     __________|                                  -1 + x                                            -1 + x                          __________          ||
 \/ -1 + 5*x  |  \    (1 + x)         /     \        \/ -1 + 5*x   / \              (1 + x)      \/ 1 + x /                \                                                                   (-1 + x)                                                                       \        \/ -1 + 5*x   / \(-1 + x)    (-1 + 5*x)                   (-1 + x)*\/ -1 + 5*x /                                   (-1 + x)                                             (-1 + x)                                                                                                                          (-1 + 5*x)                                                                                                                     (-1 + x)*\/ -1 + 5*x                                        (-1 + x)*\/ -1 + 5*x                                                          \/ -1 + 5*x                        /       \              \/ 1 + x / \\        \/ -1 + 5*x   / \           \/ -1 + 5*x /                                                                                                                  \/ -1 + 5*x           /|
4            *|------------------------ + ----------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
              |         _______                                         -1 + x                                                                                                                                                                                                                                                                                                                                                                                                                             -1 + x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          -1 + x                                                                                                          |
              \       \/ 1 + x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                8*(-1 + x)

$$\frac{4^{\sqrt{5 x - 1}} \left(\frac{x^{2} \sqrt{x + 1} \left(\left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \left(\frac{25 \log{\left(4 \right)}^{2}}{5 x - 1} - \frac{25 \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{60 \log{\left(4 \right)}}{\left(x - 1\right) \sqrt{5 x - 1}} + \frac{48}{\left(x - 1\right)^{2}}\right) + \frac{50 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \log{\left(4 \right)}^{2}}{5 x - 1} + \frac{5 \left(\frac{5 \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6}{x - 1}\right) \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{25 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{6 \left(\frac{5 \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6}{x - 1}\right) \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right)}{x - 1} - \frac{\left(\frac{5 \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6}{x - 1}\right) \left(\frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}^{2}}{5 x - 1} + \frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{60 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} + 24\right)}{x - 1} - \frac{120 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \log{\left(4 \right)}}{\left(x - 1\right) \sqrt{5 x - 1}} - \frac{15 \left(\frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}^{2}}{5 x - 1} + \frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{60 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} + 24\right) \log{\left(4 \right)}}{\left(x - 1\right) \sqrt{5 x - 1}} + \frac{84 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right)}{\left(x - 1\right)^{2}} + \frac{18 \left(\frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}^{2}}{5 x - 1} + \frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{60 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} + 24\right)}{\left(x - 1\right)^{2}} + \frac{\frac{375 \left(x - 1\right)^{3} \log{\left(4 \right)}^{2}}{\left(5 x - 1\right)^{2}} + \frac{125 \left(x - 1\right)^{3} \log{\left(4 \right)}^{3}}{\left(5 x - 1\right)^{\frac{3}{2}}} + \frac{375 \left(x - 1\right)^{3} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{5}{2}}} - \frac{450 \left(x - 1\right)^{2} \log{\left(4 \right)}^{2}}{5 x - 1} - \frac{450 \left(x - 1\right)^{2} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} + \frac{360 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 48}{\left(x - 1\right)^{2}}\right)}{x - 1} + \frac{3 x \left(\frac{x}{\sqrt{x + 1}} + 4 \sqrt{x + 1}\right) \left(\left(\frac{5 \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6}{x - 1}\right) \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) + \frac{5 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - \frac{6 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right)}{x - 1} - \frac{\frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}^{2}}{5 x - 1} + \frac{25 \left(x - 1\right)^{2} \log{\left(4 \right)}}{\left(5 x - 1\right)^{\frac{3}{2}}} - \frac{60 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} + 24}{x - 1}\right)}{x - 1} + \frac{3 \left(\frac{x^{2}}{\left(x + 1\right)^{2}} - \frac{4 x}{x + 1} + 8\right)}{\sqrt{x + 1}} + \frac{3 \left(\frac{5 \left(x - 1\right) \log{\left(4 \right)}}{\sqrt{5 x - 1}} - 6\right) \left(- \frac{x^{2}}{\left(x + 1\right)^{\frac{3}{2}}} + \frac{8 x}{\sqrt{x + 1}} + 8 \sqrt{x + 1}\right)}{x - 1}\right)}{8 \left(x - 1\right)^{3}}$$

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Derivative of (x^2sqrt(x+1))/((x-1)^3(1/4^sqrt(5*x-1)))

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Piecewise:

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Derivative of (x^2*sqrt(x+1))/((x-1)^3*(1/4^sqrt(5*x-1)))

The graph:

Piecewise:

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Derivative of (x^2sqrt(x+1))/((x-1)^3(1/4^sqrt(5*x-1)))