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

Derivative of e^x/(1+x)

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

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from to

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The solution

You have entered [src] 
   x 
  E  
-----
1 + x
exx+1\frac{e^{x}}{x + 1} 
E^x/(1 + x)
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)=exf{\left(x \right)} = e^{x} and g(x)=x+1g{\left(x \right)} = x + 1.

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

    1. The derivative of exe^{x} is itself.

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

    1. Differentiate x+1x + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10104000-2000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   x        x   
  e        e    
----- - --------
1 + x          2
        (1 + x)
exx+1−ex(x+1)2\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
(1−2x+1+2(x+1)2)exx+1\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
(1−3x+1+6(x+1)2−6(x+1)3)exx+1\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
Derivative of e^x/(1+x)
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