Derive the formula for differentiation from first principles

For a curve of f(x) against x, we can take the general point (x, f(x)) on the curve. By moving horizontally along the x-axis a distance of h, we also have the point (x+h, f(x+h)) on the curve. The gradient of the straight line between these two points is equal to the change in f(x) divided by the change in x, which (using our pair of coordinate points) is (f(x+h)-f(x))/x+h-x. This can be simplified to (f(x+h)-f(x))/h.Therefore, in the limit as h tends to 0 and the second point approaches the first along the curve, the gradient of the line tends to f'(x). This means that f'(x)=lim_h->0(f(x+h)-f(x))/h.