Why does the power rule work? If $$f(x)=x^u$$ then the derivative function will always be $$f'(x)=u*x^{u-1}$$
I've been trying to figure out why that makes sense and I can't quite get there.
I know it can be proven with limits, but I'm looking for something more basic, something I can  picture in my head.
The derivative should be the slope of the function.
If $$f(x)=x^3=x^2*x$$ then the slope should be $x^2$. But it isn't. The power rule says it's $3x^2$.
I understand that it has to do with having variables where in a more simple equation there would be a constant. I'm trying to understand how that exactly translates into the power rule.

If you know the product rule, you can derive this for when $u$ is a positive integer, which should give you a basic intuitive understanding.
Suppose, as an example, that $f(x) = x^2$. 
Equivalently, $f(x) = x*x$. By using the product rule, we have
$$f(x) = (1)(x) + (x)(1) $$
$$= 2x$$
More generally, suppose that $f(x) = x^u$. Suppose for now that $u$ is a positive integer, which allows us to expand like this:
$$f(x) = \underbrace{(x)(x)...(x)(x)}_{u\text{ terms}}$$
Using the product rule again, we can say
$$f(x) = \underbrace{\underbrace{(1)(x)...(x)(x)}_{u \text{ terms}} + (x)(1)...(x)(x) + ... + (x)(x)...(1)(x) + (x)(x)...(x)(1)}_{u\text{ terms}}$$
which simplifies to
$$f(x)=ux^{u-1}$$
