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Similar expressions
x^2/(x^2-1)

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

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

The solution

You have entered [src]

   2  
  x   
------
 2    
x  + 1

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

x^2/(x^2 + 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
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:

$\frac{- 2 x^{3} + 2 x \left(x^{2} + 1\right)}{\left(x^{2} + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

The graph:

Piecewise:

The solution