Picture of a derivative?

Question

How can the limit definition of a derivative be illustrated?

Answer 1

f'(x) = limit as h approaches 0 of (f(x+h)-f(x))/(x+h-x) = (f(x+h)-f(x))/h

For example if f(x) = x^2, then f'(x) = limit as h goes to zero of ((x+h)^2 - f(x))/h = ((x+h)^2-x^2)/h

(x^2 + 2xh + h^2 - x^2)/h = (2xh+h^2)/h = 2x +h = 2x+0 = 2x

which is the same as the short cut rule for finding derivatives of x to a power. Make the power into a coefficient and reduce the power by 1. f'(x) = 2x^(2-1) = 2x^1 = 2x