Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
[src]
x /x*cos(x) \
sin (x)*|-------- + log(sin(x))|
\ sin(x) /
$$\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)}$$
The second derivative
[src]
/ 2 2 \
x |/x*cos(x) \ 2*cos(x) x*cos (x)|
sin (x)*||-------- + log(sin(x))| - x + -------- - ---------|
|\ sin(x) / sin(x) 2 |
\ sin (x) /
$$\left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} - x - \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{x}{\left(x \right)}$$
The third derivative
[src]
/ 3 2 / 2 \ 3 \
x | /x*cos(x) \ 3*cos (x) /x*cos(x) \ | 2*cos(x) x*cos (x)| 2*x*cos (x) 2*x*cos(x)|
sin (x)*|-3 + |-------- + log(sin(x))| - --------- - 3*|-------- + log(sin(x))|*|x - -------- + ---------| + ----------- + ----------|
| \ sin(x) / 2 \ sin(x) / | sin(x) 2 | 3 sin(x) |
\ sin (x) \ sin (x) / sin (x) /
$$\left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{3} - 3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) + \frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{x}{\left(x \right)}$$