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The derivative of the function/ log(1/(x-1),exp)

Derivative of log(1/(x-1),exp)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   /  1     x\
log|-----, e |
   \x - 1    /
log⁡(1x−1)\log{\left(\frac{1}{x - 1} \right)} 
log(1/(x - 1), exp(x))
The graph
02468-8-6-4-2-1010-2010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
          /     /   /  1  \\\|          
          |     |log|-----||||          
    1     |  d  |   \x - 1/|||         x
- ----- + |-----|----------|||      x*e 
  x - 1   \dxi_2\log(xi_2) //|xi_2=e
ex∂∂ξ2log⁡(1x−1)log⁡(ξ2)∣ξ2=ex−1x−1e^{x} \left. \frac{\partial}{\partial \xi_{2}} \frac{\log{\left(\frac{1}{x - 1} \right)}}{\log{\left(\xi_{2} \right)}} \right|_{\substack{ \xi_{2}=e^{x} }} - \frac{1}{x - 1}
Simplify
The second derivative [src] 
              1      /    2\    /  1   \   /     /   /  1   \\\|          
            ------ + |1 + -|*log|------|   |     |log|------||||          
    1       -1 + x   \    x/    \-1 + x/   |  d  |   \-1 + x/|||         x
--------- + ---------------------------- + |-----|-----------|||      x*e 
        2                 2                \dxi_2\ log(xi_2) //|xi_2=e    
(-1 + x)                 x
ex∂∂ξ2log⁡(1x−1)log⁡(ξ2)∣ξ2=ex+1(x−1)2+(1+2x)log⁡(1x−1)+1x−1x2e^{x} \left. \frac{\partial}{\partial \xi_{2}} \frac{\log{\left(\frac{1}{x - 1} \right)}}{\log{\left(\xi_{2} \right)}} \right|_{\substack{ \xi_{2}=e^{x} }} + \frac{1}{\left(x - 1\right)^{2}} + \frac{\left(1 + \frac{2}{x}\right) \log{\left(\frac{1}{x - 1} \right)} + \frac{1}{x - 1}}{x^{2}}
Simplify
The third derivative [src] 
                                                                         2                                            /  1   \     /    2\    /  1   \                                   
                                                                     1 + -                                       2*log|------|   2*|1 + -|*log|------|                                   
                                                  1          1           x    /    2\    /  1   \       2             \-1 + x/     \    x/    \-1 + x/                                   
              /     /   /  1   \\\|             ------ + --------- + ------ + |1 + -|*log|------| + ---------- + ------------- + ---------------------     /  1      /    2\    /  1   \\
              |     |log|------||||             -1 + x           2   -1 + x   \    x/    \-1 + x/   x*(-1 + x)          2                  x             2*|------ + |1 + -|*log|------||
      2       |  d  |   \-1 + x/|||         x            (-1 + x)                                                      x                                   \-1 + x   \    x/    \-1 + x//
- --------- + |-----|-----------|||      x*e  - ------------------------------------------------------------------------------------------------------ + --------------------------------
          3   \dxi_2\ log(xi_2) //|xi_2=e                                                          2                                                                     2               
  (-1 + x)                                                                                        x                                                                     x
ex∂∂ξ2log⁡(1x−1)log⁡(ξ2)∣ξ2=ex−2(x−1)3+2((1+2x)log⁡(1x−1)+1x−1)x2−(1+2x)log⁡(1x−1)+1+2xx−1+1x−1+1(x−1)2+2(1+2x)log⁡(1x−1)x+2x(x−1)+2log⁡(1x−1)x2x2e^{x} \left. \frac{\partial}{\partial \xi_{2}} \frac{\log{\left(\frac{1}{x - 1} \right)}}{\log{\left(\xi_{2} \right)}} \right|_{\substack{ \xi_{2}=e^{x} }} - \frac{2}{\left(x - 1\right)^{3}} + \frac{2 \left(\left(1 + \frac{2}{x}\right) \log{\left(\frac{1}{x - 1} \right)} + \frac{1}{x - 1}\right)}{x^{2}} - \frac{\left(1 + \frac{2}{x}\right) \log{\left(\frac{1}{x - 1} \right)} + \frac{1 + \frac{2}{x}}{x - 1} + \frac{1}{x - 1} + \frac{1}{\left(x - 1\right)^{2}} + \frac{2 \left(1 + \frac{2}{x}\right) \log{\left(\frac{1}{x - 1} \right)}}{x} + \frac{2}{x \left(x - 1\right)} + \frac{2 \log{\left(\frac{1}{x - 1} \right)}}{x^{2}}}{x^{2}}
Simplify
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