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y=(x²+1)/(x²-1)

Derivative of y=(x²+1)/(x²-1)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 2    
x  + 1
------
 2    
x  - 1
$$\frac{x^{2} + 1}{x^{2} - 1}$$
(x^2 + 1)/(x^2 - 1)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Apply the power rule: goes to

      The result is:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Apply the power rule: goes to

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
             / 2    \
 2*x     2*x*\x  + 1/
------ - ------------
 2                2  
x  - 1    / 2    \   
          \x  - 1/   
$$\frac{2 x}{x^{2} - 1} - \frac{2 x \left(x^{2} + 1\right)}{\left(x^{2} - 1\right)^{2}}$$
The second derivative [src]
  /                       /          2 \\
  |              /     2\ |       4*x  ||
  |              \1 + x /*|-1 + -------||
  |         2             |           2||
  |      4*x              \     -1 + x /|
2*|1 - ------- + -----------------------|
  |          2                 2        |
  \    -1 + x            -1 + x         /
-----------------------------------------
                       2                 
                 -1 + x                  
$$\frac{2 \left(- \frac{4 x^{2}}{x^{2} - 1} + 1 + \frac{\left(x^{2} + 1\right) \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right)}{x^{2} - 1}\right)}{x^{2} - 1}$$
The third derivative [src]
     /                          /          2 \\
     |                 /     2\ |       2*x  ||
     |               2*\1 + x /*|-1 + -------||
     |          2               |           2||
     |       4*x                \     -1 + x /|
12*x*|-2 + ------- - -------------------------|
     |           2                  2         |
     \     -1 + x             -1 + x          /
-----------------------------------------------
                            2                  
                   /      2\                   
                   \-1 + x /                   
$$\frac{12 x \left(\frac{4 x^{2}}{x^{2} - 1} - 2 - \frac{2 \left(x^{2} + 1\right) \left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)}{x^{2} - 1}\right)}{\left(x^{2} - 1\right)^{2}}$$
The graph
Derivative of y=(x²+1)/(x²-1)