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The derivative of the function/ x^(lnx-1)

Derivative of x^(lnx-1)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 log(x) - 1
x
xlog⁡(x)−1x^{\log{\left(x \right)} - 1} 
x^(log(x) - 1)
Detail solution

  1. Don't know the steps in finding this derivative.

    But the derivative is

    (log⁡(x)−1)log⁡(x)−1(log⁡(log⁡(x)−1)+1)\left(\log{\left(x \right)} - 1\right)^{\log{\left(x \right)} - 1} \left(\log{\left(\log{\left(x \right)} - 1 \right)} + 1\right)

  2. Now simplify:

    (log⁡(x)−1)log⁡(x)−1(log⁡(log⁡(x)−1)+1)\left(\log{\left(x \right)} - 1\right)^{\log{\left(x \right)} - 1} \left(\log{\left(\log{\left(x \right)} - 1 \right)} + 1\right)

The answer is:

(log⁡(x)−1)log⁡(x)−1(log⁡(log⁡(x)−1)+1)\left(\log{\left(x \right)} - 1\right)^{\log{\left(x \right)} - 1} \left(\log{\left(\log{\left(x \right)} - 1 \right)} + 1\right)

The graph
02468-8-6-4-2-1010-200000100000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 log(x) - 1 /log(x) - 1   log(x)\
x          *|---------- + ------|
            \    x          x   /
xlog⁡(x)−1(log⁡(x)−1x+log⁡(x)x)x^{\log{\left(x \right)} - 1} \left(\frac{\log{\left(x \right)} - 1}{x} + \frac{\log{\left(x \right)}}{x}\right)
Simplify
The second derivative [src] 
 -1 + log(x) /                   2           \
x           *\3 + (-1 + 2*log(x))  - 2*log(x)/
----------------------------------------------
                       2                      
                      x
xlog⁡(x)−1((2log⁡(x)−1)2−2log⁡(x)+3)x2\frac{x^{\log{\left(x \right)} - 1} \left(\left(2 \log{\left(x \right)} - 1\right)^{2} - 2 \log{\left(x \right)} + 3\right)}{x^{2}}
Simplify
The third derivative [src] 
 -1 + log(x) /                    3                                               \
x           *\-8 + (-1 + 2*log(x))  + 4*log(x) - 3*(-1 + 2*log(x))*(-3 + 2*log(x))/
-----------------------------------------------------------------------------------
                                          3                                        
                                         x
xlog⁡(x)−1(−3(2log⁡(x)−3)(2log⁡(x)−1)+(2log⁡(x)−1)3+4log⁡(x)−8)x3\frac{x^{\log{\left(x \right)} - 1} \left(- 3 \left(2 \log{\left(x \right)} - 3\right) \left(2 \log{\left(x \right)} - 1\right) + \left(2 \log{\left(x \right)} - 1\right)^{3} + 4 \log{\left(x \right)} - 8\right)}{x^{3}}
Simplify
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