Section 2.9 The Derivative as a Function
In the de…nition of the derivative of a function f(x)at the point a;
f0(a) = lim
h!0
f(a+h)f(a)
h
we regarded the point aas …xed. Now we set afree.
De…nition. Let f(x)be a function of x. The derivative of fis the function,
denoted by f0;whose domain consists of all xin the domain of the function ffor
which
lim
h!0
f(x+h)f(x)
h
exists and whose value at such an xis
f0(x) = lim
h!0
f(x+h)f(x)
h:
The function f0is derived from the function fby the limiting process indicated
above. Thus the name “derivative”for f0:In geometric terms,
f0(x) = the slope of the tangent line to the graph of f
at the point (x; f (x)) :
Terminology and Notation
We say a function fis di¤erentiable on an interval Iif f0(x)exists for all x
in Iwith appropriate one-sided derivatives understood at the endpoints of Ithat
belong to I:
If y=f(x)is a function of xthen the following notations are all used to
denote the derivative of fat x:
f0(x); y0;dy
dx;df
dx;d
dxf(x); Df (x); Dxf(x)
The fractional forms stem from Leibniz as a suggestive way to remember that
derivatives stem from limits of di¤erence quotients
dy
dx = lim
x!0
y
x:
