Undergraduate Analysis pp 62-72 | Cite as

# Differentiation

Chapter

## 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*.

## Keywords

Rational Number Inverse Function Closed Interval Open Interval Chain Rule
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1983