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Understanding The Derivative
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Introductory
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Tunc Geveci
Introductory Calculus
Understanding The Derivative
Tunc Geveci
Introductory Calculus: Understanding The Derivative
Copyright © Cognella Academic Publishing 2015
www.cognella.com
All rights reserved. No part of this publication may be reproduced,
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except for brief quotations, not to exceed 250 words, without the prior
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ISBN-13: 978-1-60650-855-8 (e-book)
A Momentum Press publication
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or registered trademarks, and are used only for identification and
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Cover and interior design by S4Carlisle Publishing Services Private Ltd.,
Chennai, India
Contents
Chapter 1 The Foundation of the Derivative ................................. 1
The Derivative of a Function at a Point ......................... 1
The Derivative as a Function ....................................... 10
The Leibniz Notation .................................................. 14
Chapter 2 Using the Derivative for Powers and
Linear Combinations .............................................. 19
The Derivatives of Rational Powers of x ...................... 19
The Derivatives of Linear Combinations ..................... 29
Higher-Order Derivatives ............................................ 33
The Proof of the Power Rule for Arbitrary
Rational Powers.................................................. 35
Chapter 3 Using the Derivatives of Sine and Cosine .................... 39
The Derivatives of Sine and Cosine at 0 ...................... 39
The Derivative Functions Corresponding to
Sine and Cosine ................................................. 44
Chapter 4 Using the Derivative in Velocity and Acceleration ....... 53
Chapter 5 Local Linear Approximations ...................................... 63
The Differential .......................................................... 67
The Traditional Notation for the Differential .............. 70
The Accuracy of Local Linear Approximations ............ 74
Chapter 6 Understanding the Product and Quotient Rules .......... 77
The Quotient Rule ...................................................... 79
Chapter 7 Applying the Chain Rule ............................................. 89
A Plausibility Argument for the Chain Rule ................ 90
The Chain Rule in the Leibniz Notation ..................... 92
The Chain Rule for more than two Functions ............. 98
The Proof of the Chain Rule ..................................... 100
Chapter 8 The Problems of Related Rates .................................. 103
Chapter 9 The Intermediate Value Theorem ............................. 111
Newton’s Method ..................................................... 115
Chapter 10 Using Implicit Differentiation ................................... 125
Index .............................................................................................. 135
CHAPTER 1
The Foundation of the
Derivative
In this chapter we will introduce the concept of the derivative. The de-
rivative of a function f at a point a can be interpreted as the slope of the
tangent line to the graph of f at (a, f (a)). The tangent line is the graph
of a linear function that is the best linear approximation to f near a in a
sense that will be explained in this section.
The Derivative of a Function at a Point
In Section 1.3 we saw that the determination of the slope of a tangent
line leads to the idea of the limit. Let’s look at another example:
Example 1 Let f (x) = x2 − 2x + 4. Let’s determine the slope of the
tangent line to the graph of f at (3, f (3)) = (3, 7).
Figure 1 shows a secant line that passes through the point (3, f (3)) =
(3, 7).
y
3 h,f 3 h
f 3 h f 3
3,f 3
7 h
x
3 3 h
Figure 1
2 INTRODUCTORY CALCULUS: UNDERSTANDING THE DERIVATIVE
The slope of the secant line passing through (3, f (3)) and (3 + h, f
(3 + h)) is
( )
f(3+h)− f(3) (3+h)2 −2(3+h)+4 −7
=
h h
(7+4h+h2)−7 h(4+h)
= = =4+h.
h h
Since we expect such a secant line to be almost “tangential” to the
graph of f at (3, f (3)) if |h| is small, it is reasonable to calculate the slope
of the tangent line to the graph of f at (3, f (3)) as the limit of the slope
of the secant line as h approaches 0:
f(3+h)− f(3)
lim =lim(4+h)=4.
h→0 h h→0
Since the tangent line has slope 4 and passes through (3, f (3)), it is
the graph of the equation
y= f(3)+4(x−3)=7+4(x−3).
Figure 2 shows the graph of f and the tangent line at (3, f (3)). The
(cid:2)
picture is consistent with our intuitive notion of a tangent line.
y
20
3,f3
7
x
3 6
Figure 2: The tangent line to the graph of f at (3, f (3))
In the general case, assume that f (x) is defined for each x in an open
interval that contains the point a. If h ≠ 0 and |h| is small enough so that
f (a + h) is defined, the slope of the secant line that passes through the
points (a, f (a)) and (a + h, f (a + h)) is
f(a+h)− f(a)
.
h
THE FOUNDATION OF THE DERIVATIVE 3
y
a h,fa h
fa h fa
a,fa
h
x
a a h
Figure 3: A secant line
Since the secant line that passes through the points (a, f (a)) and (a +
h, f (a + h)) is almost “tangential” to the graph of f at (a, f (a)) if |h| is
small, we will define the slope of the tangent line to the graph of f at (a,
f (a)) as
f(a+h)− f(a)
lim .
h→0 h
The slope of a tangent line to the graph of a function can be treated
within the framework of the concept of the derivative:
Definition 1 Assume that f (x) is defined for each x in an open inter-
val that contains the point a. The derivative of f at a is
f(a+h)− f(a)
lim
h→0 h
provided that the limit exists.
We denote the derivative of f at a as f (cid:2)(a) (read “f prime at a”), so
that
f (a+h)− f (a)
f ′(a)= lim .
h→0 h
Thus, f (cid:2) (a) can be interpreted as the slope of the tangent line to
the graph of f at (a, f (a)). The tangent line to the graph of f at (a, f (a))
is the graph of the equation
= + ′ −
y f(a) f (a)(x a)
(the point-slope form of the equation of the tangent line).
4 INTRODUCTORY CALCULUS: UNDERSTANDING THE DERIVATIVE
We will refer to the ratio
f(a+h)− f(a)
h
as a difference quotient, since f (a + h) − f (a) is the difference be-
tween the values of f at a + h and a, and h is the difference between a + h
and a. Graphically, a difference quotient can be interpreted as the slope
of a secant line.
If we set x = a + h, then x approaches a as h approaches 0. Therefore,
f(a+h)− f(a) f(x)− f(a)
f ′(a)=lim =lim .
h→0 h h→0 x−a
We will favor the expression in terms of h.
Example 2 Assume that f is a linear function, so that f (x) = mx + b,
where m and b are given constants. The graph of f is a line with slope m.
Therefore, we should have f (cid:2) (a) = m at each point a. Indeed,
f(a+h)− f(a) [m(a+h)+b]−[ma+b] ma+mh+b−ma−b
= =
h h b
mh
= =m.
h
Therefore,
f(a+h)− f(a)
f ′(a)=lim =limm=m.
h→0 h h→0
(cid:2)
y
fa h
fa h fa
fa
h
a a h x
Figure 4
THE FOUNDATION OF THE DERIVATIVE 5
Example 3 Let f (x) = x3.
a) Determine f (cid:2) (2).
b) Determine the tangent line to the graph of f at (2, f (2)).
Solution
a) The relevant difference quotient is
f(2+h)− f(2) (2+h)3 −23
=
h h
23 +3(22)h+3(2)(h2)+h3 −23
=
h
12h+6h2 +h3
=
h
=12+6h+h2.
Therefore,
(2+h)3 −23
f ′(2)=lim =lim(12+6h+h2)=12.
h→0 h h→0
b) Since the tangent line to the graph of f at (2, f (2)) = (2, 8) has
slope f (cid:2) (2) = 12 and passes through (2, f (2)) = (2, 8), it is the graph of
the equation
y= f(2)+ f′(2)(x−2)=8+12(x−2).
Figure 5 displays the graph f and the tangent line at (2, f (2)).
y
16
8 2,f2
x
2 1 1 2 3
8
Figure 5
Figure 6 illustrates the effect of zooming in towards the point of
(cid:2)
contact (2, f (2)) = (2, 8) (the dashed line is the tangent line).
