Mister Exam

Derivative of (xsinx)^sin(xsinx)

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

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The solution

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          sin(x*sin(x))
(x*sin(x))             
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}}$$
(x*sin(x))^sin(x*sin(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]
          sin(x*sin(x)) /                                                  (x*cos(x) + sin(x))*sin(x*sin(x))\
(x*sin(x))             *|(x*cos(x) + sin(x))*cos(x*sin(x))*log(x*sin(x)) + ---------------------------------|
                        \                                                               x*sin(x)            /
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}} \left(\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)$$
The second derivative [src]
                        /                                                                  2                                                                                                                                                                                                           2                                                         \
          sin(x*sin(x)) |                   2 /                              sin(x*sin(x))\                       2                                                                                    (-2*cos(x) + x*sin(x))*sin(x*sin(x))   (x*cos(x) + sin(x))*sin(x*sin(x))   2*(x*cos(x) + sin(x)) *cos(x*sin(x))   (x*cos(x) + sin(x))*cos(x)*sin(x*sin(x))|
(x*sin(x))             *|(x*cos(x) + sin(x)) *|cos(x*sin(x))*log(x*sin(x)) + -------------|  - (x*cos(x) + sin(x)) *log(x*sin(x))*sin(x*sin(x)) - (-2*cos(x) + x*sin(x))*cos(x*sin(x))*log(x*sin(x)) - ------------------------------------ - --------------------------------- + ------------------------------------ - ----------------------------------------|
                        |                     \                                 x*sin(x)  /                                                                                                                          x*sin(x)                              2                                    x*sin(x)                                     2                   |
                        \                                                                                                                                                                                                                                 x *sin(x)                                                                     x*sin (x)                /
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}} \left(- \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)^{2} - \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)} \sin{\left(x \sin{\left(x \right)} \right)} - \frac{\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}}\right)$$
The third derivative [src]
                        /                                                                  3                                                                                                                                                                              /                                                                                                                                                                                                        2                                                         \                                                                                                  3                                      2                                                                                                                                                                                                                                                             2                             2                                                                                                                                  \
          sin(x*sin(x)) |                   3 /                              sin(x*sin(x))\                       3                                                                                                         /                              sin(x*sin(x))\ |                   2                                                                                    (-2*cos(x) + x*sin(x))*sin(x*sin(x))   (x*cos(x) + sin(x))*sin(x*sin(x))   2*(x*cos(x) + sin(x)) *cos(x*sin(x))   (x*cos(x) + sin(x))*cos(x)*sin(x*sin(x))|   (x*cos(x) + sin(x))*sin(x*sin(x))   (3*sin(x) + x*cos(x))*sin(x*sin(x))   3*(x*cos(x) + sin(x)) *sin(x*sin(x))   3*(x*cos(x) + sin(x)) *cos(x*sin(x))   2*(x*cos(x) + sin(x))*sin(x*sin(x))   2*(-2*cos(x) + x*sin(x))*sin(x*sin(x))                                                                              6*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*cos(x*sin(x))   3*(x*cos(x) + sin(x)) *cos(x)*cos(x*sin(x))   2*cos (x)*(x*cos(x) + sin(x))*sin(x*sin(x))   2*(-2*cos(x) + x*sin(x))*cos(x)*sin(x*sin(x))   2*(x*cos(x) + sin(x))*cos(x)*sin(x*sin(x))|
(x*sin(x))             *|(x*cos(x) + sin(x)) *|cos(x*sin(x))*log(x*sin(x)) + -------------|  - (x*cos(x) + sin(x)) *cos(x*sin(x))*log(x*sin(x)) - (3*sin(x) + x*cos(x))*cos(x*sin(x))*log(x*sin(x)) - 3*(x*cos(x) + sin(x))*|cos(x*sin(x))*log(x*sin(x)) + -------------|*|(x*cos(x) + sin(x)) *log(x*sin(x))*sin(x*sin(x)) + (-2*cos(x) + x*sin(x))*cos(x*sin(x))*log(x*sin(x)) + ------------------------------------ + --------------------------------- - ------------------------------------ + ----------------------------------------| + --------------------------------- - ----------------------------------- - ------------------------------------ - ------------------------------------ + ----------------------------------- + -------------------------------------- + 3*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*log(x*sin(x))*sin(x*sin(x)) - ---------------------------------------------------------- - ------------------------------------------- + ------------------------------------------- + --------------------------------------------- + ------------------------------------------|
                        |                     \                                 x*sin(x)  /                                                                                                                                 \                                 x*sin(x)  / |                                                                                                                      x*sin(x)                              2                                    x*sin(x)                                     2                   |                x*sin(x)                             x*sin(x)                              x*sin(x)                               2                                      3                                      2                                                                                                                             x*sin(x)                                                  2                                             3                                              2                                         2    2                   |
                        \                                                                                                                                                                                                                                                 \                                                                                                                                                           x *sin(x)                                                                     x*sin (x)                /                                                                                                                                 x *sin(x)                              x *sin(x)                              x *sin(x)                                                                                                                                                                           x*sin (x)                                     x*sin (x)                                      x*sin (x)                                     x *sin (x)                /
$$\left(x \sin{\left(x \right)}\right)^{\sin{\left(x \sin{\left(x \right)} \right)}} \left(3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \sin{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)^{3} - \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} - 3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right) \left(\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \log{\left(x \sin{\left(x \right)} \right)} \sin{\left(x \sin{\left(x \right)} \right)} + \frac{\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}}\right) - \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \sin{\left(x \right)} \right)} - \frac{6 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \right)} \cos{\left(x \sin{\left(x \right)} \right)}}{x \sin^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{x \sin^{3}{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x^{2} \sin^{2}{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}}{x^{3} \sin{\left(x \right)}}\right)$$