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Identical expressions
one / four (ln((x- one)/(x+ one)))
1 divide by 4(ln((x minus 1) divide by (x plus 1)))
one divide by four (ln((x minus one) divide by (x plus one)))
1/4lnx-1/x+1
1 divide by 4(ln((x-1) divide by (x+1)))
Similar expressions
1/4(ln((x-1)/(x-1)))
1/4(ln((x+1)/(x+1)))

The derivative of the function/ 1/4(ln((x-1)/(x+1)))

Derivative of 1/4(ln((x-1)/(x+1)))

The solution

You have entered [src]

   /x - 1\
log|-----|
   \x + 1/
----------
    4

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

log((x - 1)/(x + 1))/4

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{x - 1}{x + 1}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x - 1}{x + 1}$ :
  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:
  
  $\frac{2 \left(x + 1\right)}{\left(x - 1\right) \left(x + 1\right)^{2}}$
So, the result is: $\frac{x + 1}{2 \left(x - 1\right) \left(x + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 1/4(ln((x-1)/(x+1)))

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