Derivative of sqrt((x-1)/(x+1))

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    _______
   / x - 1 
  /  ----- 
\/   x + 1

$$\sqrt{\frac{x - 1}{x + 1}}$$

sqrt((x - 1)/(x + 1))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    _______                                 
   / x - 1          /    1         x - 1   \
  /  ----- *(x + 1)*|--------- - ----------|
\/   x + 1          |2*(x + 1)            2|
                    \            2*(x + 1) /
--------------------------------------------
                   x - 1

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

Simplify

The second derivative [src]

                           /                      -1 + x\
    ________               |                 -1 + ------|
   / -1 + x  /     -1 + x\ |  2       2           1 + x |
  /  ------ *|-1 + ------|*|----- + ------ + -----------|
\/   1 + x   \     1 + x / \1 + x   -1 + x      -1 + x  /
---------------------------------------------------------
                        4*(-1 + x)

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

Simplify

The third derivative [src]

                           /                                                                           2                     \
                           |                                              /     -1 + x\   /     -1 + x\       /     -1 + x\  |
    ________               |                                            3*|-1 + ------|   |-1 + ------|     3*|-1 + ------|  |
   / -1 + x  /     -1 + x\ |     1           1              1             \     1 + x /   \     1 + x /       \     1 + x /  |
  /  ------ *|-1 + ------|*|- -------- - --------- - ---------------- - --------------- - -------------- - ------------------|
\/   1 + x   \     1 + x / |         2           2   (1 + x)*(-1 + x)               2                2     4*(1 + x)*(-1 + x)|
                           \  (1 + x)    (-1 + x)                         4*(-1 + x)       8*(-1 + x)                        /
------------------------------------------------------------------------------------------------------------------------------
                                                            -1 + x

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

Simplify

The graph

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Derivative of sqrt((x-1)/(x+1))

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Piecewise:

The solution