Chapter 3: Derivatives
Section 3.1: Derivatives
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.
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Example: Find an equation of the tangent line to the graph of f (x) =
Example: Find an equation of the tangent line to the graph of f (x) =
2
2
at (2, 1).
x
√
x + 1 at x = 3.
Note: The derivative of f (x) at x = a can be interpreted as
• The slope of the tangent line to the graph of y = f (x) at x = a.
• The instantaneous rate of change of f (x) at x = a.
• The instantaneous velocity of f (x) at x = a.
Example: A particle moves along a straight line with equation of motion s(t) = t2 − 6t − 5,
where s is measured in meters and t in seconds. Find the particle’s velocity when t = 2.
Example: Find an equation of the tangent line to the graph of y = 3x2 − 5x at x = 2.
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Example: The graph of a function f is given below. Sketch the graph of the derivative.
(a)
(b)
(c)
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Definition: A function f is differentiable at x = a if f 0 (a) exists. A function is differentiable
on an open interval (a, b) if it is differentiable at every point in the interval.
Example: Where is f (x) = |x| differentiable?
Note: There are three ways for a function f (x) not to be differentiable at a point 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 a, then f is continuous at a.
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Example: The graph of a function f is given below. State, with reasons, where f is not
differentiable.
Example: Sketch the graph of f (x) = |x2 − 4| and determine where f is differentiable.
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Note: The derivative of f (x) at x = a can be defined as
f (a + h) − f (a)
f (x) − f (a)
= lim
.
x→a
h→0
h
x−a
f 0 (a) = lim
Example: Each of the limits below represents the derivative of some function f at some
number a. State such an f and a.
√
1+h−1
(a) lim
h→0
h
x3 − 8
x→2 x − 2
(b) lim
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