Differentiate x^2 from first principles

Differentiation is about finding gradients of functions. With straight lines we take the "rise of run" - the change in y over the change in x. With curves e.g. f(x) = x^2 we need to use the same idea, only we need to construct an infinitesimally small triangle to be able to do this.
Take an arbitrary x value x1 and another point (x1 + h) where h is a small positive number. we can construct a triangle between these two points and work out the gradient (delta y/delta x). this is (f(x1 + h) - f(x1))/((x1+h) - x1). given f(x) = x^2, this evaluates to (delta y)/(delta x) = 2x1 + h. To make the triangle infinitesimally small, we need to keep decreasing the size of h. We can take a limit to do this. As a shorthand for lim h -> 0 (delta y)/(delta x), we write dy/dx. Thus dy/dx = lim h -> 0 (2x1 + h) = 2x1. i.e. the gradient of the x^2 @ x1 is 2x1.