Chapter 4: Differentiation
Section 4.1: Formal Definition of the Derivative
Consider the curve C with equation y = f (x). What is the slope of the tangent line to the
curve C at x = a?
Example: Find an equation of the tangent line to the curve y = x2 + 2x at the point (1, 3).
1
Definition: The derivative of f (x) at x = a is
f (a + h) − f (a)
f (x) − f (a)
= lim
x→a
h→0
h
x−a
f 0 (a) = lim
provided that this limit exists. If the limit exists, then f is differentiable at x = a.
Example: Determine whether f (x) = |x| is differentiable at x = 0.
Note: There are three ways in which a function f can fail to be differentiable at x = a.
1. The graph of y = f (x) has a corner at x = a.
2. The function f (x) has a discontinuity at x = a.
3. The graph of y = f (x) has a vertical tangent line at x = a.
Theorem: If f is differentiable at x = a, then f is also continuous at x = a.
2
Example: The graph of a function f is given below. State, with reasons, where f is NOT
differentiable.
Example: Sketch the graph of y = |x2 − 4| and determine where the function is differentiable.
3
Definition: The derivative of f (x) is given by
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
provided that this limit exists. If the limit exists, then f is called differentiable.
Example: Find the derivative of f (x) = 3x + 2.
Example: Find the derivative of f (x) = 2x2 − 6x + 1.
4
Example: Consider the function f (x) =
2
.
x
(a) Find the derivative of f (x).
(b) Find an equation of the tangent line to the curve y = f (x) at (2, 1).
5
Example: Consider the function f (x) =
√
x + 1.
(a) Find the derivative of f (x).
(b) Find an equation of the tangent line to the curve y = f (x) at x = 3.
6
Example: The graph of a function f is given below. Sketch the graph of the derivative.
(a)
(b)
(c)
7
Definition: Suppose that an object moves along a straight line and the displacement of the
object at time t is given by the position function s(t). Then
1. The average velocity of the object from time t = a to t = b is
s(b) − s(a)
.
b−a
2. The instantaneous velocity of the object at time t = a is
v(a) = lim
h→0
s(a + h) − s(a)
.
h
Example: The displacement (in meters) of a object moving in a straight line is given by
s(t) = t2 − 8t + 18, where t is measured in seconds.
(a) Find the average velocity of the object on the time interval [3, 4]
(b) Find the instantaneous velocity at time t = 3.
8
Notation: If y = f (x), then some common notations for the derivative are
f 0 (x) = y 0 =
dy
df
d
=
=
f (x) = Df (x).
dx
dx
dx
Example: (Logistic Growth) Suppose that N (t) denotes the size of a population at time t
and the rate of population growth is given by
N
dN
= rN 1 −
dt
K
where r and K are positive constants. Determine the size of the population in equilibrium.
Example: (Chemical Reaction) Consider the chemical reaction
A + B −→ AB
in which the molecular reactants A and B form the molecular product AB. If x(t) denotes
the concentration of AB at time t, then the law of mass action states that
dx
= k(x − a)(x − b)
dt
where k is a positive constant and a and b denote the initial concentrations of A and B,
respectively. If k = 5, a = 8, and b = 3, for what values of x is dx/dt = 0? Interpret the
meaning of your answer.
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Definition: Suppose that f is a differentiable function. The normal line to the curve
y = f (x) at x = a is the line which is perpendicular to the tangent line and passes through
(a, f (a)).
Example: Find an equation of the normal line to the graph of y =
10
4
at (−1, −4).
x