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Graphing y =:
(x^2-1)/(x^2+1)
Integral of d{x}:
(x^2-1)/(x^2+1)
How do you in partial fractions?:
(x^2-1)/(x^2+1)
Identical expressions
(x^ two - one)/(x^ two + one)
(x squared minus 1) divide by (x squared plus 1)
(x to the power of two minus one) divide by (x to the power of two plus one)
(x2-1)/(x2+1)
x2-1/x2+1
(x²-1)/(x²+1)
(x to the power of 2-1)/(x to the power of 2+1)
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(x^2-1) divide by (x^2+1)
Similar expressions
(x^2+1)/(x^2+1)
(x^2-1)/(x^2-1)

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

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

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

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $2 x$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

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

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

Simplify

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 /

$$\frac{12 x \left(\frac{4 x^{2}}{x^{2} + 1} - \frac{2 \left(x^{2} - 1\right) \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - 2\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The graph

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

The graph:

Piecewise:

The solution