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

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

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)

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

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

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

Simplify

The graph

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

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Piecewise:

The solution