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exp(x)/(two *(x- one)*(x- one))
exponent of (x) divide by (2 multiply by (x minus 1) multiply by (x minus 1))
exponent of (x) divide by (two multiply by (x minus one) multiply by (x minus one))
exp(x)/(2(x-1)(x-1))
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exp(x) divide by (2*(x-1)*(x-1))
Similar expressions
exp(x)/(2*(x+1)*(x-1))
exp(x)/(2*(x-1)*(x+1))

The derivative of the function/ exp(x)/(2*(x-1)*(x-1))

Derivative of exp(x)/(2(x-1)(x-1))

The solution

You have entered [src]

         x       
        e        
-----------------
2*(x - 1)*(x - 1)

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

exp(x)/(((2*(x - 1))*(x - 1)))

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 $g{\left(x \right)} = \left(x - 1\right) \left(2 x - 2\right)$ .

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. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x - 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  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: $1$
  $g{\left(x \right)} = 2 x - 2$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $2 x - 2$ term by term:
    1. The derivative of the constant $-2$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result is: $2$
  The result is: $4 x - 4$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                       x
    1       x   (2 - 2*x - 2*(x - 1))*e 
----------*e  + ------------------------
         2                      4       
2*(x - 1)              4*(x - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of exp(x)/(2(x-1)(x-1))

The graph:

Piecewise:

The solution

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Derivative of exp(x)/(2*(x-1)*(x-1))

The graph:

Piecewise:

The solution

Derivative of exp(x)/(2(x-1)(x-1))