Mister Exam

Other calculators

(sin3x)^x
The derivative of the function/ (sin3x)^x

Derivative of (sin3x)^x

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   x     
sin (3*x)
$$\sin^{x}{\left(3 x \right)}$$ 
sin(3*x)^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] 
   x      /3*x*cos(3*x)                \
sin (3*x)*|------------ + log(sin(3*x))|
          \  sin(3*x)                  /
$$\left(\frac{3 x \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \log{\left(\sin{\left(3 x \right)} \right)}\right) \sin^{x}{\left(3 x \right)}$$
Simplify
The second derivative [src] 
          /                              2                             2     \
   x      |/3*x*cos(3*x)                \          6*cos(3*x)   9*x*cos (3*x)|
sin (3*x)*||------------ + log(sin(3*x))|  - 9*x + ---------- - -------------|
          |\  sin(3*x)                  /           sin(3*x)         2       |
          \                                                       sin (3*x)  /
$$\left(- 9 x - \frac{9 x \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \left(\frac{3 x \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \log{\left(\sin{\left(3 x \right)} \right)}\right)^{2} + \frac{6 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}\right) \sin^{x}{\left(3 x \right)}$$
Simplify
The third derivative [src] 
          /                                    3         2                                         /                          2     \           3                     \
   x      |      /3*x*cos(3*x)                \    27*cos (3*x)     /3*x*cos(3*x)                \ |      2*cos(3*x)   3*x*cos (3*x)|   54*x*cos (3*x)   54*x*cos(3*x)|
sin (3*x)*|-27 + |------------ + log(sin(3*x))|  - ------------ - 9*|------------ + log(sin(3*x))|*|3*x - ---------- + -------------| + -------------- + -------------|
          |      \  sin(3*x)                  /        2            \  sin(3*x)                  / |       sin(3*x)         2       |        3              sin(3*x)  |
          \                                         sin (3*x)                                      \                     sin (3*x)  /     sin (3*x)                   /
$$\left(\frac{54 x \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{54 x \cos^{3}{\left(3 x \right)}}{\sin^{3}{\left(3 x \right)}} + \left(\frac{3 x \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \log{\left(\sin{\left(3 x \right)} \right)}\right)^{3} - 9 \left(\frac{3 x \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \log{\left(\sin{\left(3 x \right)} \right)}\right) \left(3 x + \frac{3 x \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} - \frac{2 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}\right) - 27 - \frac{27 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \sin^{x}{\left(3 x \right)}$$
Simplify
The graph
Derivative of (sin3x)^x
    © Mister Exam – Calculator