What is a derivative useful for? If a derivative is the rate of change. then one should be able to find out a function changes in value. And that is what the Taylor series let us do. If you know the value of a function at a, then you can find the value of a function at x The Taylor series for a function f(x) about a point a is given by > f(x) ≈ f(a) + f'(a)(x - a) + f''(a)(x - a)^2 / 2! + f'''(a)(x - a)^3 / 3! + ... For example if f(x) = x^2, we can find f(5) using f(4) f(x) = x^2, the first few terms of the Taylor series expansion around x = 4 are: > f(x) ≈ f(4) + f'(4)(x - 4) + f''(4)(x - 4)^2 / 2! + ... Here: f(4) = 4^2 = 16 f'(x) = 2x, so f'(4) = 2\*4 = 8 f''(x) = 2, so f''(4) = 2 So the series becomes: > f(x) ≈ 16 + 8\*(x - 4) + 2\*(x - 4)^2 / 2 + higher order terms We're looking for f(5), so we plug in x = 5: > f(5) ≈ 16 + 8\*(5 - 4) + 2\*(5 - 4)^2 / 2 + higher order terms = 16 + 81 + 21^2 / 2 = 16 + 8 + 1 = 25 _third derivative of x^2 is zero, so no higher terms._