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c Amy Austin, September 23, 2015
Section 3.1: Derivatives
Definition:
(1) The Derivative of f (x) is defined to be
f (x + h) − f (x)
h→0
h
f ′ (x) = lim
(2) The Derivative of f (x) at x = a is defined to be
f (a + h) − f (a)
h→0
h
f ′ (a) = lim
EXAMPLE
√ 1: Using the limit definition of the derivative, find the derivative of
f (x) = 4x − 1.
EXAMPLE 2: Using the limit definition of the derivative, find f ′ (3) for
2
f (x) =
4x + 1
c Amy Austin, September 23, 2015
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How to interpret the derivative: f ′ (a) measures:
(a) The slope of the tangent line to the graph of f (x) at x = a
(b) The instantaneous rate of change of f (x) at x = a
(c) The instantaneous velocity at x = a.
EXAMPLE 3: For f (x) = x2 − x + 8:
(i) Find the average rate of change of f (x) over the time interval [−1, 2].
(ii) Find the instantaneous rate of change at x = −1.
(2 + h)5 − 32
represents the derivative of some function
h→0
h
f (x) at some number a. Identify f (x) and a for each limit.
EXAMPLE 4: The limit lim
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c Amy Austin, September 23, 2015
Definition: Let f (x) be a function. We say f (x) is differentiable at x = a if f ′ (a)
exists.
EXAMPLE 5: Refer to the graph below to determine where f (x) is not differentiable.
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f(x)
11
00
00
11
00
11
11
00
00
11
00
11
−5
5
1
0
0
1
−7
EXAMPLE 6: Where is f (x) = |x2 − 4| not differentiable?
What does the graph of f (x) tell us about the graph of f ′ (x)? Recall that f ′ (a)
measures the slope of f (x) at x = a, provided that f (x) is differentiable at x = a.
Therefore, if we are given the graph of f (x), we can do a rough sketch of f ′ (x) by
measuring slopes of f (x) along the curve. See the next page for examples
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c Amy Austin, September 23, 2015
EXAMPLE 7: Given the graph of f (x) below, sketch the graph of the derivative.
(i)
(0,3)
(−2,0)
(4,−2)
(ii)
b
c
a
d
e