Derivative of x/(x²-1)

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  x   
------
 2    
x  - 1

$$\frac{x}{x^{2} - 1}$$

x/(x^2 - 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               2  
  1         2*x   
------ - ---------
 2               2
x  - 1   / 2    \ 
         \x  - 1/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of x/(x²-1)

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The solution