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

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

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

The solution

You have entered [src]

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

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

E^x/(1 + x^2)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/               /         2 \        /         2 \\   
|               |      4*x  |        |      2*x  ||   
|             6*|-1 + ------|   24*x*|-1 + ------||   
|               |          2|        |          2||   
|     6*x       \     1 + x /        \     1 + x /|  x
|1 - ------ + --------------- - ------------------|*e 
|         2             2                   2     |   
|    1 + x         1 + x            /     2\      |   
\                                   \1 + x /      /   
------------------------------------------------------
                             2                        
                        1 + x

\frac{\left(- \frac{6 x}{x^{2} + 1} - \frac{24 x \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + 1 + \frac{6 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1}\right) e^{x}}{x^{2} + 1}

Simplify

The graph

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

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Piecewise:

The solution