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Integral of d{x}:
e^x/(e^x-2)
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e^x/(e^x- two)
e to the power of x divide by (e to the power of x minus 2)
e to the power of x divide by (e to the power of x minus two)
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Similar expressions
e^x/(e^x+2)

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

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

The solution

You have entered [src]

   x  
  E   
------
 x    
E  - 2

$$\frac{e^{x}}{e^{x} - 2}$$

E^x/(E^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 $g{\left(x \right)} = e^{x} - 2$ .

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 $e^{x} - 2$ term by term:
  1. The derivative of the constant $-2$ is zero.
  2. The derivative of $e^{x}$ is itself.
  The result is: $e^{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x         2*x  
  e         e     
------ - ---------
 x               2
E  - 2   / x    \ 
         \E  - 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

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

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

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

Simplify

The graph

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

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The solution