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The derivative of the function/ (2cos(x))^sin(x)

Derivative of (2cos(x))^sin(x)

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

from to

Piecewise:

The solution

You have entered [src] 
          sin(x)
(2*cos(x))
(2cos⁡(x))sin⁡(x)\left(2 \cos{\left(x \right)}\right)^{\sin{\left(x \right)}} 
(2*cos(x))^sin(x)
Detail solution

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

    But the derivative is

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

The answer is:

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

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 /                          2   \
          sin(x) |                       sin (x)|
(2*cos(x))      *|cos(x)*log(2*cos(x)) - -------|
                 \                        cos(x)/
(2cos⁡(x))sin⁡(x)(log⁡(2cos⁡(x))cos⁡(x)−sin⁡2(x)cos⁡(x))\left(2 \cos{\left(x \right)}\right)^{\sin{\left(x \right)}} \left(\log{\left(2 \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right)
Simplify
The second derivative [src] 
                 /                                2                                       \
                 |/                          2   \    /       2                   \       |
          sin(x) ||                       sin (x)|    |    sin (x)                |       |
(2*cos(x))      *||cos(x)*log(2*cos(x)) - -------|  - |3 + ------- + log(2*cos(x))|*sin(x)|
                 |\                        cos(x)/    |       2                   |       |
                 \                                    \    cos (x)                /       /
(2cos⁡(x))sin⁡(x)((log⁡(2cos⁡(x))cos⁡(x)−sin⁡2(x)cos⁡(x))2−(log⁡(2cos⁡(x))+sin⁡2(x)cos⁡2(x)+3)sin⁡(x))\left(2 \cos{\left(x \right)}\right)^{\sin{\left(x \right)}} \left(\left(\log{\left(2 \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right)^{2} - \left(\log{\left(2 \cos{\left(x \right)} \right)} + \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3\right) \sin{\left(x \right)}\right)
Simplify
The third derivative [src] 
                 /                                3                                                                                                                                    \
                 |/                          2   \                                           2           4        /                          2   \ /       2                   \       |
          sin(x) ||                       sin (x)|                                      2*sin (x)   2*sin (x)     |                       sin (x)| |    sin (x)                |       |
(2*cos(x))      *||cos(x)*log(2*cos(x)) - -------|  - 3*cos(x) - cos(x)*log(2*cos(x)) - --------- - --------- - 3*|cos(x)*log(2*cos(x)) - -------|*|3 + ------- + log(2*cos(x))|*sin(x)|
                 |\                        cos(x)/                                        cos(x)        3         \                        cos(x)/ |       2                   |       |
                 \                                                                                   cos (x)                                       \    cos (x)                /       /
(2cos⁡(x))sin⁡(x)((log⁡(2cos⁡(x))cos⁡(x)−sin⁡2(x)cos⁡(x))3−3(log⁡(2cos⁡(x))cos⁡(x)−sin⁡2(x)cos⁡(x))(log⁡(2cos⁡(x))+sin⁡2(x)cos⁡2(x)+3)sin⁡(x)−log⁡(2cos⁡(x))cos⁡(x)−2sin⁡4(x)cos⁡3(x)−2sin⁡2(x)cos⁡(x)−3cos⁡(x))\left(2 \cos{\left(x \right)}\right)^{\sin{\left(x \right)}} \left(\left(\log{\left(2 \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right)^{3} - 3 \left(\log{\left(2 \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right) \left(\log{\left(2 \cos{\left(x \right)} \right)} + \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3\right) \sin{\left(x \right)} - \log{\left(2 \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{2 \sin^{4}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)}} - 3 \cos{\left(x \right)}\right)
Simplify
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