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

Derivative of e^x/(e^x-2)

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

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The solution

You have entered [src] 
   x  
  E   
------
 x    
E  - 2
exex−2\frac{e^{x}}{e^{x} - 2} 
E^x/(E^x - 2)
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)=ex−2g{\left(x \right)} = e^{x} - 2.

    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 ex−2e^{x} - 2 term by term:

      1. The derivative of the constant −2-2 is zero.

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

      The result is: exe^{x}

    Now plug in to the quotient rule:

    (ex−2)ex−e2x(ex−2)2\frac{\left(e^{x} - 2\right) e^{x} - e^{2 x}}{\left(e^{x} - 2\right)^{2}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-100005000
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Plot the graph f'(x)
The first derivative [src] 
   x         2*x  
  e         e     
------ - ---------
 x               2
E  - 2   / x    \ 
         \E  - 2/
exex−2−e2x(ex−2)2\frac{e^{x}}{e^{x} - 2} - \frac{e^{2 x}}{\left(e^{x} - 2\right)^{2}}
Simplify
The second derivative [src] 
/              /         x \   \   
|              |      2*e  |  x|   
|              |1 - -------|*e |   
|         x    |          x|   |   
|      2*e     \    -2 + e /   |  x
|1 - ------- - ----------------|*e 
|          x             x     |   
\    -2 + e        -2 + e      /   
-----------------------------------
                    x              
              -2 + e
(−(1−2exex−2)exex−2+1−2exex−2)exex−2\frac{\left(- \frac{\left(1 - \frac{2 e^{x}}{e^{x} - 2}\right) e^{x}}{e^{x} - 2} + 1 - \frac{2 e^{x}}{e^{x} - 2}\right) e^{x}}{e^{x} - 2}
Simplify
The third derivative [src] 
/              /         x         2*x  \                        \   
|              |      6*e       6*e     |  x     /         x \   |   
|              |1 - ------- + ----------|*e      |      2*e  |  x|   
|              |          x            2|      3*|1 - -------|*e |   
|         x    |    -2 + e    /      x\ |        |          x|   |   
|      3*e     \              \-2 + e / /        \    -2 + e /   |  x
|1 - ------- - ----------------------------- - ------------------|*e 
|          x                    x                         x      |   
\    -2 + e               -2 + e                    -2 + e       /   
---------------------------------------------------------------------
                                     x                               
                               -2 + e
(−3(1−2exex−2)exex−2+1−(1−6exex−2+6e2x(ex−2)2)exex−2−3exex−2)exex−2\frac{\left(- \frac{3 \left(1 - \frac{2 e^{x}}{e^{x} - 2}\right) e^{x}}{e^{x} - 2} + 1 - \frac{\left(1 - \frac{6 e^{x}}{e^{x} - 2} + \frac{6 e^{2 x}}{\left(e^{x} - 2\right)^{2}}\right) e^{x}}{e^{x} - 2} - \frac{3 e^{x}}{e^{x} - 2}\right) e^{x}}{e^{x} - 2}
Simplify
The graph
Derivative of e^x/(e^x-2)
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