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Derivative of log
Derivative of log(e)
Derivative of log(1/x)
Derivative of log(x^-1)
Identical expressions
one /4ln(x²- one)/(x²+ one)
1 divide by 4ln(x² minus 1) divide by (x² plus 1)
one divide by 4ln(x² minus one) divide by (x² plus one)
1/4lnx²-1/x²+1
1 divide by 4ln(x²-1) divide by (x²+1)
Similar expressions
1/4ln(x²+1)/(x²+1)
1/4ln(x²-1)/(x²-1)

The derivative of the function/ 1/4ln(x²-1)/(x²+1)

Derivative of 1/4ln(x²-1)/(x²+1)

The solution

You have entered [src]

/   / 2    \\
|log\x  - 1/|
|-----------|
\     4     /
-------------
     2       
    x  + 1

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

(log(x^2 - 1)/4)/(x^2 + 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x^{2} - 1 \right)}$ and $g{\left(x \right)} = 4 x^{2} + 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{2} - 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 1\right)$ :
  1. Differentiate $x^{2} - 1$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $-1$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{2 x}{x^{2} - 1}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $4 x^{2} + 4$ term by term:
  1. The derivative of the constant $4$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $8 x$
  The result is: $8 x$
Now plug in to the quotient rule:

$\frac{- 8 x \log{\left(x^{2} - 1 \right)} + \frac{2 x \left(4 x^{2} + 4\right)}{x^{2} - 1}}{\left(4 x^{2} + 4\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                           / 2    \
         x            x*log\x  - 1/
------------------- - -------------
  / 2    \ / 2    \              2 
2*\x  + 1/*\x  - 1/      / 2    \  
                       2*\x  + 1/

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

Simplify

The second derivative [src]

            2    /         2 \                                  
         2*x     |      4*x  |    /      2\                     
  -1 + -------   |-1 + ------|*log\-1 + x /                     
             2   |          2|                          2       
       -1 + x    \     1 + x /                       2*x        
- ------------ + -------------------------- - ------------------
    /      2\              /     2\           /     2\ /      2\
  2*\-1 + x /            2*\1 + x /           \1 + x /*\-1 + x /
----------------------------------------------------------------
                                  2                             
                             1 + x

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

Simplify

The third derivative [src]

  /          2      /         2 \                   /          2 \       /         2 \  \
  |       4*x       |      2*x  |    /      2\      |       2*x  |       |      4*x  |  |
  |-3 + -------   6*|-1 + ------|*log\-1 + x /    3*|-1 + -------|     3*|-1 + ------|  |
  |           2     |          2|                   |           2|       |          2|  |
  |     -1 + x      \     1 + x /                   \     -1 + x /       \     1 + x /  |
x*|------------ - ---------------------------- + ------------------ + ------------------|
  |          2                     2             /     2\ /      2\   /     2\ /      2\|
  | /      2\              /     2\              \1 + x /*\-1 + x /   \1 + x /*\-1 + x /|
  \ \-1 + x /              \1 + x /                                                     /
-----------------------------------------------------------------------------------------
                                               2                                         
                                          1 + x

$$\frac{x \left(- \frac{6 \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right) \log{\left(x^{2} - 1 \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{3 \left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} + \frac{3 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} + \frac{\frac{4 x^{2}}{x^{2} - 1} - 3}{\left(x^{2} - 1\right)^{2}}\right)}{x^{2} + 1}$$

Simplify

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Derivative of 1/4ln(x²-1)/(x²+1)

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