Note about Differentiation

A tangent line to a function at a point is the line that best approximates the function
at that point better than any other line.
The slope of the function at a given point is the slope of the tangent line to the
function at that point.
The derivative of f at x = a is the slope, m, of the function f at the point x = a (if m
exists), denoted by f0(a) = m. All other notations:
y0, dy
dx , df
dx , d
dxf(x), Dxy, Dxf(x).
The function f(x) is dierentiable at a point x0 if f0(x0) exists. If a function is
dierentiable at all points in its domain (i.e. f0(x) is dened for all x in the domain), then
we consider f0(x) as a function and call it the derivative of f(x).
The derivative of f that we have been talking about is called the rst derivative. Now,
we dene the second derivative of a function to be the derivative of f0, denoted by f00(x)
or d2f
dx2 (= d
dx
?? d
dxf)


.
Example 1: Given f(x) = c where c is a constant. Then f0(x) = 0 because the slope of
the function at each point is zero.
Example 2: If f(x) = 2 ?? 3x , then the derivative f0(x) = 2 because the slope of the
function at each point is 2.
Example 3: Given f(x) = jxj. We have
f0(x) =
(
??1 if x < 0
1 if x > 0
:
However, f0(0) is not dened because there is no unique tangent line to f(x) at x = 0.
The following is a table of derivatives of some basic functions:
f(x) f0(x)
c 0
mx + c m
xa axa??1
ex ex
ln x 1
x
1
Rules of Dierentiation:
1. (f  g)0 = f0  g0
2. (c
f)0 = cf0
3. (Product Rule) (f
g)0 = f0g + fg0
4. (Quotient Rule)

f
g
0
= f0g ?? fg0
g2 (where g(x) 6= 0)
5. (Chain Rule) (f  g)0 = (f(g(x)))0 = f0(g(x))
g0(x)
The equation of the tangent line to the function at point x = x0 is:
y ?? f(x0) = f0(x0)(x ?? x0)
Theorem (The Extreme-Value Theorem for Continuous Functions)
If f is continuous at every point of a closed interval I, then f assumes both an absolute
maximum value value M and an absolute minimum value m somewhere in I.
Denition
A point in the domain of a function f at which f