Mister Exam

Derivative of sin(x)^x

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

The graph:

from to

Piecewise:

The solution

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

    But the derivative is


The answer is:

The graph
The first derivative [src]
   x    /x*cos(x)              \
sin (x)*|-------- + log(sin(x))|
        \ sin(x)               /
$$\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)}$$
The second derivative [src]
        /                        2                       2   \
   x    |/x*cos(x)              \        2*cos(x)   x*cos (x)|
sin (x)*||-------- + log(sin(x))|  - x + -------- - ---------|
        |\ sin(x)               /         sin(x)        2    |
        \                                            sin (x) /
$$\left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} - x - \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{x}{\left(x \right)}$$
The third derivative [src]
        /                             3        2                                 /                    2   \          3                \
   x    |     /x*cos(x)              \    3*cos (x)     /x*cos(x)              \ |    2*cos(x)   x*cos (x)|   2*x*cos (x)   2*x*cos(x)|
sin (x)*|-3 + |-------- + log(sin(x))|  - --------- - 3*|-------- + log(sin(x))|*|x - -------- + ---------| + ----------- + ----------|
        |     \ sin(x)               /        2         \ sin(x)               / |     sin(x)        2    |        3          sin(x)  |
        \                                  sin (x)                               \                sin (x) /     sin (x)               /
$$\left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{3} - 3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) + \frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{x}{\left(x \right)}$$
The graph
Derivative of sin(x)^x