Differentiation

Abstract

Let f be a function defined on an interval having more than one point, say I. Let x ∈ I. We shall say that f is differentiable at x if the limit of the Newton quotient

$$\mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}$$

exists. It is understood that the limit is taken for x + h = I. Thus if x is, say, a left end point of the interval, we consider only values of h > 0. We see no reason to limit ourselves to open intervals. If f is differentiable at x, it is obviously continuous at x. If the above limit exists, we call it the derivative of f at x, and denote it by f′(x). If f is differentiable at every point of I, then f′is a function on I.