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Derivative of:
Derivative of (x^2+1)^3
Derivative of sin(cosx)
Derivative of ln(-x)
Derivative of i*n*sin(x)
Identical expressions
ln((one +x)/(one -x))^(one / two)
ln((1 plus x) divide by (1 minus x)) to the power of (1 divide by 2)
ln((one plus x) divide by (one minus x)) to the power of (one divide by two)
ln((1+x)/(1-x))(1/2)
ln1+x/1-x1/2
ln1+x/1-x^1/2
ln((1+x) divide by (1-x))^(1 divide by 2)
Similar expressions
ln((1-x)/(1-x))^(1/2)
ln((1+x)/(1+x))^(1/2)

The derivative of the function/ ln((1+x)/(1-x))^(1/2)

Derivative of ln((1+x)/(1-x))^(1/2)

The solution

You have entered [src]

    ____________
   /    /1 + x\ 
  /  log|-----| 
\/      \1 - x/

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

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /  1      1 + x  \
(1 - x)*|----- + --------|
        |1 - x          2|
        \        (1 - x) /
--------------------------
              ____________
             /    /1 + x\ 
2*(1 + x)*  /  log|-----| 
          \/      \1 - x/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln((1+x)/(1-x))^(1/2)

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The solution