I have come across the function $\sin^{x}(x)$ (or a in a different notation$(\sin(x)^{x})$ while practicing calculating its the first derivative. I then plot it in Desmos since I don't know how to graph this by hands yet. It seems that the graphing device suggests this function is discontinuous between the interval $[\pi ,2\pi ]$. The graph is also peculiar between the interval $[0 ,\pi ]$, while replicating itself for any interval $(2\pi ,3\pi )$, $(3\pi ,4\pi )$, $(5\pi ,6\pi )$, etc.
The behaviour of this function on the entire interval $(-\infty ,0)$ is also interesting. It doesn't look anywhere similar to $[0,+∞)$.
The first derivative of this function is $\sin^{x}(x)+\ln(\sin(x))+\frac{x\cos(x)}{\sin(x)}$. As with most other complex transcendental functions, this is not a simple derivative to work with. Could you show me some strategies to explore and graph this function manually. I would like to particularly find the red circle point found on this graph:
I have also tried to plot this on Wolfram and its shape is similar. Is this an accurate shape at all or is the graphing device errorneous somehow?
The canonical way to define a power $a^b$ is to do $$ a^b:=e^{b\log a}. $$ This works great, but the problem is that it doesn't make sense when $a<0$. And that's the state of things. There is no natural way to define arbitrary powers of negative numbers (think $(-1)^{1/2}$ for an easy example).
So your function is not defined whenever $\sin x<0$, which is all the intervals $((2k-1)\pi,2k\pi)$. That's why your graph has so many gaps.
On the left axis, you have negative powers of $\sin x$, and that's why you get the asymptotic behaviour towards $x=-k\pi$, $k\in\mathbb N$.
I don't think you can expect to find the roots of the derivative analytically. At best, you could try Newton's method.