Mister Exam

Derivative of y=(sin2x)^cos2x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   cos(2*x)     
sin        (2*x)
$$\sin^{\cos{\left(2 x \right)}}{\left(2 x \right)}$$
sin(2*x)^cos(2*x)
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]
                 /                                 2     \
   cos(2*x)      |                            2*cos (2*x)|
sin        (2*x)*|-2*log(sin(2*x))*sin(2*x) + -----------|
                 \                              sin(2*x) /
$$\left(- 2 \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} + \frac{2 \cos^{2}{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right) \sin^{\cos{\left(2 x \right)}}{\left(2 x \right)}$$
The second derivative [src]
                   /                                    2                                           \
                   |/                            2     \    /       2                     \         |
     cos(2*x)      ||                         cos (2*x)|    |    cos (2*x)                |         |
4*sin        (2*x)*||log(sin(2*x))*sin(2*x) - ---------|  - |3 + --------- + log(sin(2*x))|*cos(2*x)|
                   |\                          sin(2*x)/    |       2                     |         |
                   \                                        \    sin (2*x)                /         /
$$4 \left(\left(\log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} - \frac{\cos^{2}{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right)^{2} - \left(\log{\left(\sin{\left(2 x \right)} \right)} + 3 + \frac{\cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)}\right) \sin^{\cos{\left(2 x \right)}}{\left(2 x \right)}$$
The third derivative [src]
                   /                                      3                                                                                                                                                    \
                   |  /                            2     \                                               2             4          /                            2     \ /       2                     \         |
     cos(2*x)      |  |                         cos (2*x)|                                          2*cos (2*x)   2*cos (2*x)     |                         cos (2*x)| |    cos (2*x)                |         |
8*sin        (2*x)*|- |log(sin(2*x))*sin(2*x) - ---------|  + 3*sin(2*x) + log(sin(2*x))*sin(2*x) + ----------- + ----------- + 3*|log(sin(2*x))*sin(2*x) - ---------|*|3 + --------- + log(sin(2*x))|*cos(2*x)|
                   |  \                          sin(2*x)/                                            sin(2*x)        3           \                          sin(2*x)/ |       2                     |         |
                   \                                                                                               sin (2*x)                                           \    sin (2*x)                /         /
$$8 \left(- \left(\log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} - \frac{\cos^{2}{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right)^{3} + 3 \left(\log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} - \frac{\cos^{2}{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right) \left(\log{\left(\sin{\left(2 x \right)} \right)} + 3 + \frac{\cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)} + \log{\left(\sin{\left(2 x \right)} \right)} \sin{\left(2 x \right)} + 3 \sin{\left(2 x \right)} + \frac{2 \cos^{2}{\left(2 x \right)}}{\sin{\left(2 x \right)}} + \frac{2 \cos^{4}{\left(2 x \right)}}{\sin^{3}{\left(2 x \right)}}\right) \sin^{\cos{\left(2 x \right)}}{\left(2 x \right)}$$
The graph
Derivative of y=(sin2x)^cos2x