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e*exp(x*loge(x- one))
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Similar expressions
e*exp(x*loge(x+1))

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

Derivative of eexp(xloge(x-1))

The solution

You have entered [src]

   x*log(x - 1)
   ------------
        / 1\   
     log\e /   
E*e

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

E*exp((x*log(x - 1))/log(exp(1)))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

                        x*log(x - 1)
                        ------------
                             / 1\   
  /  x               \    log\e /   
E*|----- + log(x - 1)|*e            
  \x - 1             /              
------------------------------------
                 / 1\               
              log\e /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of eexp(xloge(x-1))

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Piecewise:

The solution

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Derivative of e*exp(x*loge(x-1))

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Piecewise:

The solution

Derivative of eexp(xloge(x-1))