Chapter 6
Differentiable Functions
Sections 6.1-6.3 develop the standard introductory material
on derivatives and differentiability. Section 6 .4 covers impor-
tant mean-value type theorems and their applications. Sec-
t ion 6.5 is a basic introduction to Taylor's theorem and its
applications. Section 6.6 is an optional section on L'Hopital's
rule. It can be assigned as a project for independent study.In this chapter, you will finally feel that you are studying "calculus,'' because
our focus will be on derivatives and differentiability.6.1 The Derivative and Differentiability
Definition 6.1.1 Suppose f : D(f) ~JR and Xo is an interior point of D(f).
Then f is differentiable at xo if the limit lim f(x) - f(xo) exists (i.e., is
x-->xo x - xo
finite). If this limit exists, we call it the derivative of f at xo, and denote it
f '(xo).
Thus, f'(x 0 ) = lim f(x) - f(xo) if this limit exists.
X-->Xo x - XoTheorem 6.1.2 Every "linear"^1 function f(x) = ax+b is differentiable at all
Xo E JR, and f'(xo) =a.- The t erm "linear" is time-honored but incorrect. "Affine" would be more correct. In linear
algebra, "linear" h as a more restrictive definition.
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