We have come a long way and finally are about to study calculus. Many of you might have
taken some courses in the past where you learned a number of formulas to calculate the derivatives
and integrals of certain functions. The purpose of this course, however, is not to memorize these
formulas mindlessly. Rather, our goals are to understand the mathematical concepts underlying
such formulas and to develop a solid understanding of calculus. This should not be too challenging
given that we are now armed with the knowledge of sequential and functional limits.
1 Derivatives
First, we start with the familiar definition of a derivative.
Definition 1 Let f : X 7! R be a function and c 2 X be an accumulation point of X. Then, the
derivative is defined as
f0(c) = lim
x!c
f(x) ? f(c)
x ? c
.
We say f is differentiable at c if this limit exists. If such a limit exists at all c 2 X, then we say
f is differentiable (on X).
Before moving on, it is important to develop a geometric understanding of this definition. That is,
the derivative represents a rate of change of the function. Also, make sure that you can apply the
definition above to the following examples.
Example 1 Are the following functions differentiable at c 2 R?
1. f : R 7! R where f(x) = 2x + 1.
2. f : R 7! R where f(x) = xn with n 2 N.
3. f : R 7! R where f(x) = |x|.
The last example suggests a connection between differentiability and continuity. Indeed, if a function
is differentiable at a certain point, then it is also continuous at that point.
Theorem 1 (Differentiability and Continuity) Let f : X 7! R be a differentiable function at
c 2 X. Then, f is continuous at c.
We note that although a function must be continuous if it is differentiable, its derivative might not
be continuous. That is, the derivative of a derivative, called the second derivative, may not.
function whose derivative is differentiable is said to be twice differentiable. The same principle
applies to the third, fourth, . . . , derivatives.
There is important connection between derivatives and sequential limits. The next theorem
immediately follows from Theorem 1 of the previous chapter.
Theorem 2 (Derivatives and Sequential Limits) Let f : X 7! R be a function and c be an
accumulation point of X. Then, f is differentiable at c if and only if for every sequence {xn}1
n=1 of
X with xn 6= c for all n 2 N and limn!1 xn = c, the sequence
converges to f0(c).
Now, we prove the algebraic rules concerning derivatives
Theorem 3 (Algebraic Operations of Derivatives) Let f, g : X 7! R be differentiable at c 2
X and X R. Then,
1. (kf)0(c) = kf0(c) for all k 2 R,
2. (f + g)0(c) = f0(c) + g0(c),
3. (fg)0(c) = f0(c)g(c) + f(c)g0(c) (Product Rule),
(c) = f0(c)g(c)?f(c)g0(c)
g(c)2 for g(c) 6= 0 (Quotient Rule.
Definition 2 (Exponential and Logarithmic Functions) The exponential function f : R 7!
(0,1) is defined as f(x) = ex where e = limn!1
??
1 + 1
n
n = 2.71828 . . .. The inverse of the
exponential function, f?1 : (0,1) 7! R is called the (natural) logarithmic function and is denoted
by log x.
That is, if y = ex, then x = log y. The important properties of exponential and logarithmic
functions are: exey = ex+y, (ex)y = exy, log xy = y log x, log x + log y = log xy, and log x ? log y =
log x
y . You may occasionally encounter a log function whose base is different from e. For example,
g : (0,1) 7! R with g(x) = log10 x is called the common logarithmic function. That is, if y = 10x
then x = log10 y. Now, we derive the derivative of the natural logarithmic function.
Theorem 5 (Derivative of Logarithmic Function) Consider f : (0,1) 7! R with f(x) =
log(x). Then, f0(c) = 1
c for all c 2 (0,1).
2 The Mean Value Theorem and Its Applications
Derivatives are often used to solve the optimization problems of functions where the goal is to find
a point where an objective function attains its maximum or minimum. We first define the concept
of local (or relative) extremum.
Definition 3 (Local Extremum) Let f : X 7! R be a function with X R. A point c 2 X
is a local maximum (minimum) if there exists a neighborhood Q of c such that if f(x) f(c)
(f(x) f(c)) for any x 2 Q \ X.
Sometimes, one use the word global (or absolute) extremum to distinguish extremum from local
extremum. Obviously, a local extremum is not guaranteed to be a global extremum because the
former is an extremum only in a neighborhood. The next theorem shows that f0(c) = 0 is a
necessary condition for a point c being an extremum.
Theorem 6 (Interior Extremum Theorem) Let f : X 7! R be differentiable on (a, b) X
with a < b. If f attains a maximum or minimum value at some point c 2 (a, b), then f0(c) = 0.
Note that as we see below, f0(c) = 0 is NOT a sufficient condition for an extremum, local or global.
Now, recall the Extremum Value Theorem from the previous chapter which states that continuous
functions on compact sets always attain maximum and minimum values. By combining this with
Theorem 6, we have the following result.
Theorem 7 (Rolle’s Theorem) Let f : [a, b] 7! R be a function that is continuous on [a, b] and
differentiable on (a, b) with a < b. If f(a) = f(b), there exists a point c 2 (a, b) such that f0(c) = 0.
It turns out that this is a special case of an even more general result, which is the key theorem of
this section! After proving it, we also collect important implications of the Mean Value Theorem.
Theorem 8 (Mean Value Theorem) Let f : [a, b] 7! R be a function that is continuous on
[a, b] and differentiable on (a, b) with a < b. Then, there exists a point c 2 (a, b) such that