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Identical expressions
sqrt((2x+ three)/(2x- three))
square root of ((2x plus 3) divide by (2x minus 3))
square root of ((2x plus three) divide by (2x minus three))
√((2x+3)/(2x-3))
sqrt2x+3/2x-3
sqrt((2x+3) divide by (2x-3))
Similar expressions
sqrt((2x+3)/(2x+3))
sqrt((2x-3)/(2x-3))
sqrt((2*x+3)/(2*x-3))*ctg(3*x^2+5)

The derivative of the function/ sqrt((2x+3)/(2x-3))

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

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

Let $u = \frac{2 x + 3}{2 x - 3}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 3}{2 x - 3}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 x + 3$ and $g{\left(x \right)} = 2 x - 3$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $2 x + 3$ term by term:
    1. The derivative of the constant $3$ 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: $x$ goes to $1$
      So, the result is: $2$
    The result is: $2$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $2 x - 3$ term by term:
    1. The derivative of the constant $-3$ 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: $x$ goes to $1$
      So, the result is: $2$
    The result is: $2$
  Now plug in to the quotient rule:
  
  $- \frac{12}{\left(2 x - 3\right)^{2}}$
The result of the chain rule is:

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

$- \frac{6}{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right)^{2}}$

The answer is:

$- \frac{6}{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(2 x - 3\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

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

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

Simplify

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

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

Simplify

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

$$\frac{\sqrt{\frac{2 x + 3}{2 x - 3}} \left(1 - \frac{2 x + 3}{2 x - 3}\right) \left(\frac{\left(1 - \frac{2 x + 3}{2 x - 3}\right)^{2}}{\left(2 x + 3\right)^{2}} - \frac{6 \left(1 - \frac{2 x + 3}{2 x - 3}\right)}{\left(2 x + 3\right)^{2}} - \frac{6 \left(1 - \frac{2 x + 3}{2 x - 3}\right)}{\left(2 x - 3\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x + 3\right)^{2}} + \frac{8}{\left(2 x - 3\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x - 3\right)^{2}}\right)}{2 x + 3}$$

Simplify

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

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