Differentiation

Abstract

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

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

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.