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

Derivative of x/(x^2+1)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  x   
------
 2    
x  + 1
xx2+1\frac{x}{x^{2} + 1} 
x/(x^2 + 1)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=xf{\left(x \right)} = x and g(x)=x2+1g{\left(x \right)} = x^{2} + 1.

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

    1. Apply the power rule: xx goes to 11

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x2+1x^{2} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: x2x^{2} goes to 2x2 x

      The result is: 2x2 x

    Now plug in to the quotient rule:

    1−x2(x2+1)2\frac{1 - x^{2}}{\left(x^{2} + 1\right)^{2}}

The answer is:

1−x2(x2+1)2\frac{1 - x^{2}}{\left(x^{2} + 1\right)^{2}}

The graph
02468-8-6-4-2-10102-2
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/
−2x2(x2+1)2+1x2+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 /
2x(4x2x2+1−3)(x2+1)2\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 /
6(−4x2(2x2x2+1−1)x2+1+4x2x2+1−1)(x2+1)2\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
Derivative of x/(x^2+1)
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