Derivative of a Function

Definition 1: The derivative of the function f
with respect to the variable x is the function
f’ whose value at x is
f x  x   f x 
f ' ( x)  lim
x 0
x
Provided the limit exists.

Differentiate.
1. f x   x
3

Differentiate.
2. f x   x

There are many ways to denote the derivative
of a function y = f(x). Besides f’(x), the most
common notations are these:
◦ y’
“y prime”
Nice and brief, but
does not name the
independent variable.
◦ dy/dx
“dy dx” or
Names both variables
“the derivative
of y with
respect to x”
◦ df/dx
◦ d/dx f(x)
“df dx” or
“the derivative
of f with
respect to x”
Emphasizes the
function name
“d dx of f at x”
Emphasizes the
or “the derivative idea that
of f at x”
differentiation is
an operation
performed on f.

Use the definition to find the derivative of the
given function at the indicated point.
𝑓 𝑥 = 𝑥 3 + 𝑥, 𝑎 = 1

Graph the derivative of the function f whose
graph is shown below. Discuss the behavior
of f in terms of the signs and values of f’.

Sketch the graph of a function f that has the
following properties:
◦ f(0) = 0;
◦ the graph of f’, the derivative of f, is as shown
in the figure below;
◦ f is continuous for all x.

Suppose 30 people are in a room. What is the
probability that two of them share the same
birthday? Ignore the year of birth. People in Probabilit
Room (x)
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
y
0
0.027
0.117
0.253
0.411
0.569
0.706
0.814
0.891
0.941
0.970
0.986
0.994
0.998
0.999

A function y = f(x) is differentiable on a
closed interval [a, b] if it has a derivative at
every interior point of the interval, and if the
limits
f a  x   f a 
lim
x 0 
x
f b  x   f b 
lim 
x 0
x
[the right - hand derivative at a]
[the left - hand derivative at b]
exist at the endpoints.

Show that the following function has lefthand and right-hand derivatives at x = 0, but
no derivative there.
x2 , x  0
y
2 x, x  0

Using one-sided derivatives, show that the
function
𝑥 3,
𝑓 𝑥 =
3𝑥,
𝑥≤1
𝑥>1
does not have a derivative at x = 1.