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2*x/(x^2+1)

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

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

The graph:

from to

Piecewise:

The solution

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

    and .

    To find :

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: goes to

      So, 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         4*x   
------ - ---------
 2               2
x  + 1   / 2    \ 
         \x  + 1/ 
$$- \frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2}{x^{2} + 1}$$
The second derivative [src]
    /         2 \
    |      4*x  |
4*x*|-3 + ------|
    |          2|
    \     1 + x /
-----------------
            2    
    /     2\     
    \1 + x /     
$$\frac{4 x \left(\frac{4 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{2}}$$
The third derivative [src]
   /                   /         2 \\
   |                 2 |      2*x  ||
   |              4*x *|-1 + ------||
   |         2         |          2||
   |      4*x          \     1 + x /|
12*|-1 + ------ - ------------------|
   |          2              2      |
   \     1 + x          1 + x       /
-------------------------------------
                      2              
              /     2\               
              \1 + x /               
$$\frac{12 \left(- \frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} + \frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}}$$
The graph
Derivative of 2*x/(x^2+1)