Page 79 - Calculus Demystified

CHAPTER 2
66
2.4 The Derivative Foundations of Calculus
Suppose that f is a function whose domain contains the interval (a, b). Let c be a
point of (a, b). If the limit
f(c + h) − f(c)
lim (∗)
h→0 h
exists then we say that f is differentiable at c and we call the limit the derivative
of f at c.

EXAMPLE 2.10
2
Isthe function f(x) = x + x differentiable at x = 2? If it is, calculate the
derivative.
SOLUTION
We calculate the limit (∗), with the role of c played by 2:
2
2
f(2 + h) − f(2) [(2 + h) + (2 + h)]−[2 + 2]
lim = lim
h→0 h h→0 h
2
[(4 + 4h + h ) + (2 + h)]−[6]
= lim
h→0 h
5h + h 2
= lim
h→0 h
= lim 5 + h
h→0
= 5.
We see that the required limit (∗) exists, and that it equals 5. Thus the function
2
f(x) = x + x is differentiable at x = 2, and the value of the derivative is 5.
Math Note: When the derivative of a function f exists at a point c, then we denote
the derivative either by f (c) or by (d/dx)f (c) = (df /dx)(c). In some contexts

˙
(e.g., physics) the notation f(c) is used. In the last example, we calculated that
f (2) = 5.

The importance of the derivative is two-fold: it can be interpreted as rate of
change and it can be interpreted as the slope. Let us now consider both of these
ideas.
Suppose that ϕ(t) represents the position (in inches or feet or some other standard
unit) of a moving body at time t. At time 0 the body is at ϕ(0), at time 3 the body is
at ϕ(3), and so forth. Imagine that we want to determine the instantaneous velocity
of the body at time t = c. What could this mean? One reasonable interpretation
is that we can calculate the average velocity over a small interval at c, and let the