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

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $-1$
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:

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

The answer is:

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

The graph

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/

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

Simplify

The second derivative [src]

    /        2 \
    |     4*x  |
2*x*|3 - ------|
    |         2|
    \    1 + x /
----------------
           2    
   /     2\     
   \1 + x /

$$\frac{2 x \left(- \frac{4 x^{2}}{x^{2} + 1} + 3\right)}{\left(x^{2} + 1\right)^{2}}$$

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 /

$$\frac{6 \left(\frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{4 x^{2}}{x^{2} + 1} + 1\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The graph

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Identical expressions

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

The graph:

Piecewise:

The solution