"Let us begin with the simple expression y = x2 . ..Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. ...the enlarged y will be equal to the square of the enlarged x. Writing this down we have y + dy = (x + dx)2. Doing the squaring [see Ch.2, Fig.2 above] we get y + dy = x^2 + 2x\cdot dx + (dx)^2. ...dx2 will mean a little bit of a little bit of x; that is... a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have: y + dy = x^2 + 2x\cdot dx. Now y = x2; so let us subtract this from the equation and we have left dy = 2x\cdot dx. Dividing across by dx, we find dy/dx = 2x."