Undergraduate Analysis pp 66-77 | Cite as

# Differentiation

Chapter

## Abstract

Let exists. It is understood that the limit is taken for

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

*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*.## Keywords

Inverse Function Closed Interval Open Interval Chain Rule Lower Form## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer Science+Business Media New York 1997