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Derivative of sqrt((2x+3)/(2x-3))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
    _________
   / 2*x + 3 
  /  ------- 
\/   2*x - 3 
$$\sqrt{\frac{2 x + 3}{2 x - 3}}$$
sqrt((2*x + 3)/(2*x - 3))
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Apply the quotient rule, which is:

      and .

      To find :

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      To find :

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
    _________                                 
   / 2*x + 3  /   1       2*x + 3  \          
  /  ------- *|------- - ----------|*(2*x - 3)
\/   2*x - 3  |2*x - 3            2|          
              \          (2*x - 3) /          
----------------------------------------------
                   2*x + 3                    
$$\frac{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right) \left(\frac{1}{2 x - 3} - \frac{2 x + 3}{\left(2 x - 3\right)^{2}}\right)}{2 x + 3}$$
The second derivative [src]
                              /                           3 + 2*x \
    __________                |                       1 - --------|
   / 3 + 2*x   /    3 + 2*x \ |     2          2          -3 + 2*x|
  /  -------- *|1 - --------|*|- -------- - ------- + ------------|
\/   -3 + 2*x  \    -3 + 2*x/ \  -3 + 2*x   3 + 2*x     3 + 2*x   /
-------------------------------------------------------------------
                              3 + 2*x                              
$$\frac{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(\frac{1 - \frac{2 x + 3}{2 x - 3}}{2 x + 3} - \frac{2}{2 x + 3} - \frac{2}{2 x - 3}\right)}{2 x + 3}$$
The third derivative [src]
                              /                                         2                                                                 \
                              |                           /    3 + 2*x \      /    3 + 2*x \                              /    3 + 2*x \  |
    __________                |                           |1 - --------|    6*|1 - --------|                            6*|1 - --------|  |
   / 3 + 2*x   /    3 + 2*x \ |     8            8        \    -3 + 2*x/      \    -3 + 2*x/            8                 \    -3 + 2*x/  |
  /  -------- *|1 - --------|*|----------- + ---------- + --------------- - ---------------- + -------------------- - --------------------|
\/   -3 + 2*x  \    -3 + 2*x/ |          2            2               2                 2      (-3 + 2*x)*(3 + 2*x)   (-3 + 2*x)*(3 + 2*x)|
                              \(-3 + 2*x)    (3 + 2*x)       (3 + 2*x)         (3 + 2*x)                                                  /
-------------------------------------------------------------------------------------------------------------------------------------------
                                                                  3 + 2*x                                                                  
$$\frac{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(\frac{\left(1 - \frac{2 x + 3}{2 x - 3}\right)^{2}}{\left(2 x + 3\right)^{2}} - \frac{6 \left(1 - \frac{2 x + 3}{2 x - 3}\right)}{\left(2 x + 3\right)^{2}} - \frac{6 \left(1 - \frac{2 x + 3}{2 x - 3}\right)}{\left(2 x - 3\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x + 3\right)^{2}} + \frac{8}{\left(2 x - 3\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x - 3\right)^{2}}\right)}{2 x + 3}$$