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Graphing y =:
e^x/(1+x)
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e^x/(1+x)
Limit of the function:
e^x/(1+x)
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Similar expressions
e^x/(1-x)

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

Derivative of e^x/(1+x)

The solution

You have entered [src]

   x 
  E  
-----
1 + x

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

E^x/(1 + x)

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 $g{\left(x \right)} = x + 1$ .

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 $x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x        x   
  e        e    
----- - --------
1 + x          2
        (1 + x)

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

Simplify

The second derivative [src]

/      2        2    \  x
|1 - ----- + --------|*e 
|    1 + x          2|   
\            (1 + x) /   
-------------------------
          1 + x

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

Simplify

The third derivative [src]

/       6         3        6    \  x
|1 - -------- - ----- + --------|*e 
|           3   1 + x          2|   
\    (1 + x)            (1 + x) /   
------------------------------------
               1 + x

$$\frac{\left(1 - \frac{3}{x + 1} + \frac{6}{\left(x + 1\right)^{2}} - \frac{6}{\left(x + 1\right)^{3}}\right) e^{x}}{x + 1}$$

Simplify

The graph

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Derivative of e^x/(1+x)

The graph:

Piecewise:

The solution