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  • (x^2*sinx)^(cosx)
The derivative of the function/ (x^2*sin(x))^(cos(x))

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

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

The graph:

from to

Piecewise:

The solution

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

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

    But the derivative is

The answer is:

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