Mister Exam

Derivative of y=x^(e^ctg(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 / cot(x)\
 \e      /
x         
$$x^{e^{\cot{\left(x \right)}}}$$
  / / cot(x)\\
d | \e      /|
--\x         /
dx            
$$\frac{d}{d x} x^{e^{\cot{\left(x \right)}}}$$
Detail solution
  1. Don't know the steps in finding this derivative.

    But the derivative is


The answer is:

The first derivative [src]
 / cot(x)\ / cot(x)                                \
 \e      / |e         /        2   \  cot(x)       |
x         *|------- + \-1 - cot (x)/*e      *log(x)|
           \   x                                   /
$$x^{e^{\cot{\left(x \right)}}} \left(\left(- \cot^{2}{\left(x \right)} - 1\right) e^{\cot{\left(x \right)}} \log{\left(x \right)} + \frac{e^{\cot{\left(x \right)}}}{x}\right)$$
The second derivative [src]
 / cot(x)\ /                    2                                      2             /       2   \                                \        
 \e      / |  1    /       2   \           /  1   /       2   \       \   cot(x)   2*\1 + cot (x)/     /       2   \              |  cot(x)
x         *|- -- + \1 + cot (x)/ *log(x) + |- - + \1 + cot (x)/*log(x)| *e       - --------------- + 2*\1 + cot (x)/*cot(x)*log(x)|*e      
           |   2                           \  x                       /                   x                                       |        
           \  x                                                                                                                   /        
$$x^{e^{\cot{\left(x \right)}}} \left(\left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} - \frac{1}{x}\right)^{2} e^{\cot{\left(x \right)}} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \cot{\left(x \right)} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) e^{\cot{\left(x \right)}}$$
The third derivative [src]
            /                                                                                        2                                                                                                                                                                                                                                                                 \        
  / cot(x)\ |                    3                                      3               /       2   \      /       2   \                  2            /       2   \                                         /                    2            /       2   \                                \                                                           2              |        
  \e      / |  2    /       2   \           /  1   /       2   \       \   2*cot(x)   3*\1 + cot (x)/    3*\1 + cot (x)/     /       2   \           6*\1 + cot (x)/*cot(x)     /  1   /       2   \       \ |  1    /       2   \           2*\1 + cot (x)/     /       2   \              |  cot(x)        2    /       2   \            /       2   \               |  cot(x)
-x         *|- -- + \1 + cot (x)/ *log(x) + |- - + \1 + cot (x)/*log(x)| *e         - ---------------- - --------------- + 2*\1 + cot (x)/ *log(x) - ---------------------- + 3*|- - + \1 + cot (x)/*log(x)|*|- -- + \1 + cot (x)/ *log(x) - --------------- + 2*\1 + cot (x)/*cot(x)*log(x)|*e       + 4*cot (x)*\1 + cot (x)/*log(x) + 6*\1 + cot (x)/ *cot(x)*log(x)|*e      
            |   3                           \  x                       /                     x                   2                                             x                \  x                       / |   2                                  x                                       |                                                                          |        
            \  x                                                                                                x                                                                                            \  x                                                                           /                                                                          /        
$$- x^{e^{\cot{\left(x \right)}}} \left(\left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} - \frac{1}{x}\right)^{3} e^{2 \cot{\left(x \right)}} + \left(\cot^{2}{\left(x \right)} + 1\right)^{3} \log{\left(x \right)} + 6 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} \cot{\left(x \right)} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \cot^{2}{\left(x \right)} + 3 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} - \frac{1}{x}\right) \left(\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \cot{\left(x \right)} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) e^{\cot{\left(x \right)}} + 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} - \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{x} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x} - \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2}} - \frac{2}{x^{3}}\right) e^{\cot{\left(x \right)}}$$