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The derivative of the function/ (log2sinx)^(4x)

Derivative of (log2sinx)^(4x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   4*x          
log   (2*sin(x))
log⁡(2sin⁡(x))4x\log{\left(2 \sin{\left(x \right)} \right)}^{4 x} 
log(2*sin(x))^(4*x)
Detail solution

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

    But the derivative is

    (4x)4x(log⁡(4x)+1)\left(4 x\right)^{4 x} \left(\log{\left(4 x \right)} + 1\right)

The answer is:

(4x)4x(log⁡(4x)+1)\left(4 x\right)^{4 x} \left(\log{\left(4 x \right)} + 1\right)

The graph
02468-8-6-4-2-1010-1e615e60
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   4*x           /                            4*x*cos(x)     \
log   (2*sin(x))*|4*log(log(2*sin(x))) + --------------------|
                 \                       log(2*sin(x))*sin(x)/
(4xcos⁡(x)log⁡(2sin⁡(x))sin⁡(x)+4log⁡(log⁡(2sin⁡(x))))log⁡(2sin⁡(x))4x\left(\frac{4 x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + 4 \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right) \log{\left(2 \sin{\left(x \right)} \right)}^{4 x}
Simplify
The second derivative [src] 
                   /                                                                     2                 2         \
                   |                                                     2*cos(x)   x*cos (x)         x*cos (x)      |
                   |                                                 x - -------- + --------- + ---------------------|
                   |                                             2        sin(x)        2                        2   |
     4*x           |  /      x*cos(x)                           \                    sin (x)    log(2*sin(x))*sin (x)|
4*log   (2*sin(x))*|4*|-------------------- + log(log(2*sin(x)))|  - ------------------------------------------------|
                   \  \log(2*sin(x))*sin(x)                     /                     log(2*sin(x))                  /
4(4(xcos⁡(x)log⁡(2sin⁡(x))sin⁡(x)+log⁡(log⁡(2sin⁡(x))))2−x+xcos⁡2(x)sin⁡2(x)+xcos⁡2(x)log⁡(2sin⁡(x))sin⁡2(x)−2cos⁡(x)sin⁡(x)log⁡(2sin⁡(x)))log⁡(2sin⁡(x))4x4 \left(4 \left(\frac{x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right)^{2} - \frac{x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \cos^{2}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\log{\left(2 \sin{\left(x \right)} \right)}}\right) \log{\left(2 \sin{\left(x \right)} \right)}^{4 x}
Simplify
The third derivative [src] 
                   /                                                            2                 2                   3                               3                        3                                                                                 /                    2                 2         \\
                   |                                                       3*cos (x)         3*cos (x)         2*x*cos (x)   2*x*cos(x)        2*x*cos (x)              3*x*cos (x)             3*x*cos(x)           /      x*cos(x)                           \ |    2*cos(x)   x*cos (x)         x*cos (x)      ||
                   |                                                  -3 - --------- - --------------------- + ----------- + ---------- + ---------------------- + --------------------- + --------------------   12*|-------------------- + log(log(2*sin(x)))|*|x - -------- + --------- + ---------------------||
                   |                                              3            2                        2           3          sin(x)        2              3                       3      log(2*sin(x))*sin(x)      \log(2*sin(x))*sin(x)                     / |     sin(x)        2                        2   ||
     4*x           |   /      x*cos(x)                           \          sin (x)    log(2*sin(x))*sin (x)     sin (x)                  log (2*sin(x))*sin (x)   log(2*sin(x))*sin (x)                                                                         \                sin (x)    log(2*sin(x))*sin (x)/|
4*log   (2*sin(x))*|16*|-------------------- + log(log(2*sin(x)))|  + ----------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------|
                   \   \log(2*sin(x))*sin(x)                     /                                                                  log(2*sin(x))                                                                                                           log(2*sin(x))                                          /
4(16(xcos⁡(x)log⁡(2sin⁡(x))sin⁡(x)+log⁡(log⁡(2sin⁡(x))))3−12(xcos⁡(x)log⁡(2sin⁡(x))sin⁡(x)+log⁡(log⁡(2sin⁡(x))))(x+xcos⁡2(x)sin⁡2(x)+xcos⁡2(x)log⁡(2sin⁡(x))sin⁡2(x)−2cos⁡(x)sin⁡(x))log⁡(2sin⁡(x))+2xcos⁡(x)sin⁡(x)+2xcos⁡3(x)sin⁡3(x)+3xcos⁡(x)log⁡(2sin⁡(x))sin⁡(x)+3xcos⁡3(x)log⁡(2sin⁡(x))sin⁡3(x)+2xcos⁡3(x)log⁡(2sin⁡(x))2sin⁡3(x)−3−3cos⁡2(x)sin⁡2(x)−3cos⁡2(x)log⁡(2sin⁡(x))sin⁡2(x)log⁡(2sin⁡(x)))log⁡(2sin⁡(x))4x4 \left(16 \left(\frac{x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right)^{3} - \frac{12 \left(\frac{x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \cos^{2}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{\log{\left(2 \sin{\left(x \right)} \right)}} + \frac{\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{3 x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \frac{3 x \cos^{3}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{3}{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{3 \cos^{2}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}}{\log{\left(2 \sin{\left(x \right)} \right)}}\right) \log{\left(2 \sin{\left(x \right)} \right)}^{4 x}
Simplify
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