Math 151
Section 3.1
Derivatives
Definition of a Derivative The derivative of f ( x ) at x = a, denoted f !( a) , is given by
f !( a) = lim
f ( x ) # f ( a)
x"a
x#a
= lim
h"0
f ( a + h) # f ( a)
h
Alternate Notation Other ways to represent the derivative of y = f ( x) include:
f ! , y! ,
dy
d
f ( x) .
, and
dx
dx
Example: Find the derivative of f ( x ) at the given value of x.
A. f ( x) =
2
at x = 2
x +5
B. f ( x ) = x at x = 3
C. f ( x ) = x 2 for any x
Math 151
Example: The limits below represent the derivative of some function f ( x ) at x = a. Identify f ( x )
and a for each limit.
A. lim
x!3!
cos x +1
x " 3!
(1+ h)
2013
B. lim
h!0
"1
h
Interpretations of the Derivative f !( a) represents:
•
The slope of the tangent line to the graph of f ( x ) at x = a.
•
The instantaneous rate of change of f ( x ) at x = a.
•
The instantaneous velocity at x = a.
Example: For f ( x) =
2
, find the equation of the tangent line at x = 2.
x +5
Math 151
Differentiable If f !( a) exists, f ( x ) is said to be differentiable at x = a. f ( x ) is differentiable on
an open interval (a, b) if it is differentiable at every number in the interval.
Theorem If f ( x ) is differentiable at x = a, then f ( x ) is continuous at x = a.
Example: Sketch the graph of f ( x) = 2x ! 4 and use this graph to find f !( x) . Identify the values
where f ( x ) is not continuous and where it is not differentiable.
Example: Sketch a possible graph of the derivative for each of the following graphs.