Mister Exam

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

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

from to

### The solution

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

and .

To find :

1. The derivative of is itself.

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 graph
The first derivative [src]
   x            x
e        2*x*e
------ - ---------
2               2
x  + 1   / 2    \
\x  + 1/ 
$$- \frac{2 x e^{x}}{\left(x^{2} + 1\right)^{2}} + \frac{e^{x}}{x^{2} + 1}$$
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              
$$\frac{\left(- \frac{4 x}{x^{2} + 1} + 1 + \frac{2 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1}\right) e^{x}}{x^{2} + 1}$$
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                         
$$\frac{\left(- \frac{6 x}{x^{2} + 1} - \frac{24 x \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + 1 + \frac{6 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1}\right) e^{x}}{x^{2} + 1}$$
The graph