Strange behavior of $\sin(x^3)$ and $\tan(x^3)$ I noticed this behavior a long time ago and never really figured this out but if you take the $\sin$ or $\cos$ or $\tan$ etc. of a cubic polynomial you get a very strange and erratic behavior. It seems to have no easily explainable pattern and I'm not sure why. The function is still smooth but the successive derivatives grow ever larger. My best guess is that because the derivatives keep growing larger and larger but are still periodic you get this behavior because the every more violent derivatives keep canceling each other out.
Is this an example of a simple chaotic system? What features of these functions make it behave so erratically. Can I do this with other kinds of periodic functions?
In short, why does this function behave so erratically?
 A: When plotting rapid oscillating functions on graphical calculators one often run into problems with the calculator using a too coarse sampling of the point making the plot look very erratic (the fairly large pixel-size on these calculators does not help either). As OP mentioned below; his plots looks very different from the one I present here so this is likely the source of the apparent "chaotic" behavior.

The period of $\sin(x)$ is $2\pi$ so starting at a given value $x$ we have that $\sin(x^3)$ completes one oscillation when reaching $x + \Delta x$ where 
$$(x+\Delta x)^3 - x^3 = 2\pi \implies\Delta x \approx \frac{2\pi}{3x^2}$$
for large $x$. We see that $\Delta x$ becomes smaller and smaller as $x$ grows so the function oscillates more and more rapidly as we increase $x$. This in turn implies very large derivatives for large $x$. We can also show this by computing the derivatives exactly
$$\frac{d}{dx}\sin(x^3) = 3x^2\cos(x^3)$$
The graph of this will be rapid oscillations contained within the envelope of $\pm 3x^2$:
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You can get this behavior with any type of periodic function $f(x)$ by considering $f(g(x))$ where $g(x)$ grows faster than $x$. For example taking $g(x)=e^x$ gives even more rapid oscillations than for $g(x) = x^3$. Here is a comparison of $\sin(e^{2x})$ and $\sin(x^3)$:
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