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The derivative of the function/ y=sin^nx*cosnx

Derivative of y=sin^nx*cosnx

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

The graph:

from to

Piecewise:

The solution

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

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

    But the derivative is

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

The answer is:

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

The first derivative [src] 
                 /                          2                       \
   n*x           |                       x*n *cos(cos(n*x))*sin(n*x)|
sin   (cos(n*x))*|n*log(sin(cos(n*x))) - ---------------------------|
                 \                              sin(cos(n*x))       /
(−n2xsin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))+nlog⁡(sin⁡(cos⁡(nx))))sin⁡nx(cos⁡(nx))\left(- \frac{n^{2} x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + n \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right) \sin^{n x}{\left(\cos{\left(n x \right)} \right)}
Simplify
The second derivative [src] 
                    /                                                  2                                                     2              2                                  \
 2    n*x           |/                      n*x*cos(cos(n*x))*sin(n*x)\           2        2*cos(cos(n*x))*sin(n*x)   n*x*cos (cos(n*x))*sin (n*x)   n*x*cos(n*x)*cos(cos(n*x))|
n *sin   (cos(n*x))*||-log(sin(cos(n*x))) + --------------------------|  - n*x*sin (n*x) - ------------------------ - ---------------------------- - --------------------------|
                    |\                            sin(cos(n*x))       /                         sin(cos(n*x))                   2                          sin(cos(n*x))       |
                    \                                                                                                        sin (cos(n*x))                                    /
n2(−nxsin⁡2(nx)−nxsin⁡2(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))−nxcos⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))+(nxsin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−log⁡(sin⁡(cos⁡(nx))))2−2sin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx)))sin⁡nx(cos⁡(nx))n^{2} \left(- n x \sin^{2}{\left(n x \right)} - \frac{n x \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \frac{n x \cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + \left(\frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right)^{2} - \frac{2 \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) \sin^{n x}{\left(\cos{\left(n x \right)} \right)}
Simplify
The third derivative [src] 
                    /                                                    3                                                                      /                                                  2              2                                  \        2              2                                                                                                   3              3                 3                               2                            \
 3    n*x           |  /                      n*x*cos(cos(n*x))*sin(n*x)\         2          /                      n*x*cos(cos(n*x))*sin(n*x)\ |       2        2*cos(cos(n*x))*sin(n*x)   n*x*cos (cos(n*x))*sin (n*x)   n*x*cos(n*x)*cos(cos(n*x))|   3*cos (cos(n*x))*sin (n*x)   3*cos(n*x)*cos(cos(n*x))                             n*x*cos(cos(n*x))*sin(n*x)   2*n*x*cos (cos(n*x))*sin (n*x)   2*n*x*sin (n*x)*cos(cos(n*x))   3*n*x*cos (cos(n*x))*cos(n*x)*sin(n*x)|
n *sin   (cos(n*x))*|- |-log(sin(cos(n*x))) + --------------------------|  - 3*sin (n*x) + 3*|-log(sin(cos(n*x))) + --------------------------|*|n*x*sin (n*x) + ------------------------ + ---------------------------- + --------------------------| - -------------------------- - ------------------------ - 3*n*x*cos(n*x)*sin(n*x) + -------------------------- - ------------------------------ - ----------------------------- - --------------------------------------|
                    |  \                            sin(cos(n*x))       /                    \                            sin(cos(n*x))       / |                     sin(cos(n*x))                   2                          sin(cos(n*x))       |            2                        sin(cos(n*x))                                         sin(cos(n*x))                     3                             sin(cos(n*x))                          2                      |
                    \                                                                                                                           \                                                  sin (cos(n*x))                                    /         sin (cos(n*x))                                                                                                   sin (cos(n*x))                                                       sin (cos(n*x))            /
n3(−2nxsin⁡3(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−2nxsin⁡3(nx)cos⁡3(cos⁡(nx))sin⁡3(cos⁡(nx))−3nxsin⁡(nx)cos⁡(nx)+nxsin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−3nxsin⁡(nx)cos⁡(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))−(nxsin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−log⁡(sin⁡(cos⁡(nx))))3+3(nxsin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−log⁡(sin⁡(cos⁡(nx))))(nxsin⁡2(nx)+nxsin⁡2(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))+nxcos⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))+2sin⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx)))−3sin⁡2(nx)−3sin⁡2(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))−3cos⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx)))sin⁡nx(cos⁡(nx))n^{3} \left(- \frac{2 n x \sin^{3}{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \frac{2 n x \sin^{3}{\left(n x \right)} \cos^{3}{\left(\cos{\left(n x \right)} \right)}}{\sin^{3}{\left(\cos{\left(n x \right)} \right)}} - 3 n x \sin{\left(n x \right)} \cos{\left(n x \right)} + \frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \frac{3 n x \sin{\left(n x \right)} \cos{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \left(\frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right)^{3} + 3 \left(\frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right) \left(n x \sin^{2}{\left(n x \right)} + \frac{n x \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} + \frac{n x \cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + \frac{2 \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) - 3 \sin^{2}{\left(n x \right)} - \frac{3 \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \frac{3 \cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) \sin^{n x}{\left(\cos{\left(n x \right)} \right)}
Simplify
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