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The derivative of the function/ (-x*e^x+e^x-1)/(1-e^x)^2

Derivative of (-x*e^x+e^x-1)/(1-e^x)^2

The solution

You have entered [src]

    x    x    
-x*e  + e  - 1
--------------
          2   
  /     x\    
  \1 - e /

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

  /    x    x    \
d |-x*e  + e  - 1|
--|--------------|
dx|          2   |
  |  /     x\    |
  \  \1 - e /    /

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

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = - x e^{x} + e^{x} - 1$ and $g{\left(x \right)} = \left(1 - e^{x}\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- x e^{x} + e^{x} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of $e^{x}$ is itself.
      The result is: $x e^{x} + e^{x}$
    So, the result is: $- x e^{x} - e^{x}$
  3. The derivative of $e^{x}$ is itself.
  The result is: $- x e^{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 - e^{x}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - e^{x}\right)$ :
  1. Differentiate $1 - e^{x}$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of $e^{x}$ is itself.
      So, the result is: $- e^{x}$
    The result is: $- e^{x}$
  The result of the chain rule is:
  
  $- \left(2 - 2 e^{x}\right) e^{x}$
Now plug in to the quotient rule:

$\frac{- x \left(1 - e^{x}\right)^{2} e^{x} + \left(2 - 2 e^{x}\right) \left(- x e^{x} + e^{x} - 1\right) e^{x}}{\left(1 - e^{x}\right)^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x    x      x     /    x    x    \  x
e  - e  - x*e    2*\-x*e  + e  - 1/*e 
-------------- + ---------------------
          2                    3      
  /     x\             /     x\       
  \1 - e /             \1 - e /

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

Simplify

The second derivative [src]

/           /         x \                           \   
|           |      3*e  | /        x    x\          |   
|         2*|1 - -------|*\-1 - x*e  + e /          |   
|           |          x|                          x|   
|           \    -1 + e /                     4*x*e |  x
|-1 - x - -------------------------------- + -------|*e 
|                           x                      x|   
\                     -1 + e                 -1 + e /   
--------------------------------------------------------
                                2                       
                       /      x\                        
                       \-1 + e /

$$\frac{\left(- x + \frac{4 x e^{x}}{e^{x} - 1} - \frac{2 \cdot \left(1 - \frac{3 e^{x}}{e^{x} - 1}\right) \left(- x e^{x} + e^{x} - 1\right)}{e^{x} - 1} - 1\right) e^{x}}{\left(e^{x} - 1\right)^{2}}$$

Simplify

The third derivative [src]

/           /         x         2*x  \                                                       \   
|           |      9*e      12*e     | /        x    x\                      /         x \   |   
|         2*|1 - ------- + ----------|*\-1 - x*e  + e /                      |      3*e  |  x|   
|           |          x            2|                                   6*x*|1 - -------|*e |   
|           |    -1 + e    /      x\ |                               x       |          x|   |   
|           \              \-1 + e / /                    6*(1 + x)*e        \    -1 + e /   |  x
|-2 - x - --------------------------------------------- + ------------ + --------------------|*e 
|                                  x                              x                  x       |   
\                            -1 + e                         -1 + e             -1 + e        /   
-------------------------------------------------------------------------------------------------
                                                     2                                           
                                            /      x\                                            
                                            \-1 + e /

$$\frac{\left(\frac{6 x \left(1 - \frac{3 e^{x}}{e^{x} - 1}\right) e^{x}}{e^{x} - 1} - x + \frac{6 \left(x + 1\right) e^{x}}{e^{x} - 1} - \frac{2 \cdot \left(1 - \frac{9 e^{x}}{e^{x} - 1} + \frac{12 e^{2 x}}{\left(e^{x} - 1\right)^{2}}\right) \left(- x e^{x} + e^{x} - 1\right)}{e^{x} - 1} - 2\right) e^{x}}{\left(e^{x} - 1\right)^{2}}$$

Simplify

The graph

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Derivative of (-x*e^x+e^x-1)/(1-e^x)^2

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