I've been using it for a while but still don't really understand why it works.
For integer exponents greater or equal to 2, its easy to intuitively understand it using the geometric interpretation of the product rule: http://web.mit.edu/wwmath/calculus/differentiation/products.html
But could someone give me an intuition about why it works (not a proof)? An intuition that extends to non-integer and negative powers?
On
I would not call the product rule thing an intuition, it is more of a proof.
If you want an intuitive explanation for the integer case, you can for example consider $r^n$ as the volume of an $n$-ball of radius $r$ (up to a constant). As you change the radius, the volume changes and the derivative of $r^n$ tells you how fast it changes. Now it should be obvious that the larger the surface of tthe sphere, the faster the volume will grow, and the surface is $r^{n-1}$ (again, up to a constant factor)
Do you genuinely understand how or why $x'=1$? If so, then, through an effort of imagination, try to picture this x, which you regard as mono-dimensional, as being in reality a multi-dimensional entity itself, and write it as a product of n equal quantities, namely $x=y^n\iff y=\sqrt[n]x$ . Now, apply the integer-dimensional product rule, which you genuinely seem to grasp, to x, but this time viewing it as a composite n-dimensional object, and y as the new unit. What would we get ? $$x'=(y^n)'=n\cdot y^{n-1}\cdot y'\iff y'=\dfrac{x'}{n\cdot y^{n-1}}\iff\Big(\sqrt[n]x\Big)'=\dfrac1{n\cdot\Big(\sqrt[n]x\Big)^{n-1}}$$ For instance, we could represent various three-dimensional volumes using mono-dimensional line segments, e.g., $1$ cm $\equiv$ $1$ m$^3$, or $1$ inch $\equiv$ $1$ gallon, etc. So just because we use a mono-dimensional representation for something, that does not mean that that certain something is uncompounded.