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Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25
Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25
198 Chapter 2 • The Derivative and Its Properties Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e x and y = ln x 195
Clear explanations supported by = 5 · 3x = 15x 2

d
3
2
3
x
x
Suppose f (x) = a , where a > 0 and a �= 1. The derivative of f is
(a) f (x) = 5 · x , so f (x) = 5
�
dx
1
1
1 x
x
x
h
x + h
− a
2
1
f (x + h) − f (x) a (b) g(u) =− · u , so g (u) =− · a d u =− · 2u =−u
2
� a · a −
rigorous examples. = lim 2 = lim 2 du 2
f (x) = lim
�
h→0 h h→0 h ↑ h→0 h d
4
2
4
4 2
4 3
3
h
x
x + h
�
= a · a
(c) u(x) = π x , so u (x) = π · dx x = π · 3x = 3π x
a
↑
h h π is a constant
x a − 1 x a − 1
= lim a · = a · lim
h→0 h h→0 h
↑ NOW WORK Problem 31 and AP Practice Problem 10.
R
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x
Factor out a .
Support in everyday language.
h
a − 1
provided lim
exists.
Use In Words to translate the complex 3 Differentiate the Sum and the Difference of Two Functions
h
h→0
formulas, theorems, proofs, rules, and We can find the derivative of a function that is the sum of two functions whose
derivatives are known by adding the derivatives of each function.
definitions into everyday language to x
Three observations about the derivative of f (x) = a are significant:
ease your understanding of them. h THEOREM Sum Rule
h
0
• f (0) = a lim a − 1 = lim a − 1 . If two functions f and g are differentiable and if F(x) = f (x) + g(x), then F is
�
h
h
h→0
h→0
IN WORDS The derivative of the sum of two differentiable and
differentiable functions equals the sum of d
x
x
x
their derivatives. That is, ( f + g) � = f � + g � .
�
• f (x) is a multiple of a . In fact, a = f (0) · a . F (x) = f (x) + g (x)
�
�
�
�
dx
• If f (0) exists, then f (x) exists, and the domain of f is the same
Proof If F(x) = f (x) + g(x), then as that of
�
�
�
f (x) = a , all real numbers.
x
F(x + h) − F(x) = [ f (x + h) + g(x + h)] − [ f (x) + g(x)]
Sullivan 05 apcalc4e 45342 ch03 234 283 3pp July 13, 2023 9:41
= [ f (x + h) − f (x)] + [g(x + h) − g(x)]
So, the derivative of F is
The slope of the tangent line to the graph of f (x) = a x at the point (0, 1)
238
Chapter 3 • The Derivative of Composite, Implicit, and Inverse Functions
h
a − 1 F (x) = lim [ f (x + h) − f (x)] + [g(x + h) − g(x)]
is f (0) = lim , and the value of this limit depends h→0 on the base a. In Section P.5,
�
�
NEED TO REVIEW? The number e is h→0 h h • If u = u(x) is a differentiable function,
discussed in Section P.5, pp. 46–47. the number e was defined as that number for which the slope of the tangent g(x + h) − g(x) The limit of a sum is
f (x + h) − f (x)
+ lim line to the
= lim
h
h→0
h→0
the sum of the limits. d
x
graph of y = a at the point (0, 1) equals 1. That is, if f (x) = e , then f (0) = 1 so that h d sin u(x) = cos u(x) du sec u(x) = sec u(x) tan u(x) du
x
�
�
= f (x) + g (x) dx dx dx dx
�
y
d
f(x) e x e − 1 In Leibniz notation, the Sum Rule takes the form du d csc u(x) =−csc u(x) cot u(x) du
h
cos u(x) =−sin u(x)
= 1
4 lim NEED TO REVIEW? The derivatives of the dx dx dx dx
h
y x 1 h→0 trigonometric functions are discussed on d d d d du d du
Access to practice. x pp. 218 and 219 in Section 2.5. dx [ f (x) + g(x)] = dx dx tan u(x) g(x) 2 dx dx cot u(x) =−csc u(x) dx
f (x) +
2
dx = sec u(x)
Information-rich
Figure 28 shows f (x) = e
and the tangent line y = x + 1 with slope 1 at the
After reading through an EXAMPLE ,
2 point (0, 1). examples.
try the NOW WORK Practice Problems EXAMPLE 4 Differentiating the Sum of Two Functions R
Problem 41 and AP Practice Problems 3, 4, 10, 12, and 15.
NOW WORK
d
d
x
x
x
x
x
x
2 · e
a = f (0) · a , if f (x) = e , then
Since
e = f (0) · e = 1
�
�
(0, 1) and AP ® Review Problems at the x Find the derivative of f (x) = 3x + 8. = e . Worked EXAMPLES
dx
dx
provide step-by-step
end of the section to master the Solution EXAMPLE 3 Finding an Equation of a Tangent Line
2 2 4 x Here f is the sum of 3x and 8. So, we begin by using the Sum Rule. 4x instruction. Look for
2
x
concepts. THEOREM Derivative of the Exponential Function y = e Find an equation of the tangent line to the graph of y = 5e at the point (0, 5).
Figure 28 x d d d d annotations in blue
The derivative of the exponential function y = e is f (x) = (3x + 8) = Solution 8 = 3 x + 0 = 3 · 2x = 6x
2
2
2
�
(3x ) +
dx ↑ dx dx ↑ dx ↑ that show you what
The slope of the tangent line to the graph of y = f (x) at the point (0, 5) is f (0).
Sum Rule Constant Multiple Simple �
d x x Rule Power Rule formula or reasoning is
�
y = e = e (1) � d (5e ) = 5 d 4x 4x d (4x) = 5e · 4 = 20e 4x
4x
4x
dx f (x) = e = 5e · involved in solving the
y
y � 5e 4x
NOW WORK Problem 7 and AP Practice Problem 6. dx ↑ dx ↑ dx
R
Constant
d
dx problem.
20 y � 20x � 5 Multiple Rule u = 4x; dx e u = e u du
EXAMPLE 7 Differentiating an Expression Involving y = e x
0
�
CALC CLIP m tan = f (0) = 20e = 20. Using the point-slope form of a line,
10
x
3
Find the derivative of f (x) = 4e + x . (0, 5) Calculus in our world.
y − 5 = 20(x − 0)
y − y 0 = m tan (x − x 0 ).
y = 20x + 5 EXAMPLES examples show how
Applied
Solution �1 �0.5 0.5 1 x
calculus is beneficial and relevant to a wide
3
x
The function f is the sum of 4e and x . Then The graph of y = 5e 4x and the line y = 20x + 5 are shown in Figure 1.
Figure 1
d d d d variety of fields and endeavors.
3
� x 3 x x = 4 x 2 x 2 R
f (x) = (4e + x ) = (4e ) + e + 3x = 4e + 3x NOW WORK Problem 77 and AP Practice Problems 5, 7, and 16.
dx ↑ dx dx ↑ dx ↑
Sum Rule Constant Multiple Rule; Use (1).Use (1).
Constant Multiple Rule;
Sum Rule
Simple Power Rule EXAMPLE 4 Application: Carbon-14 Dating
Simple Power Rule
tice
Problem
NOW WORK Problem 25 and AP Practice Problem 4.4. All carbon on Earth contains some carbon-14, which is radioactive and exists in a fixed
Prac
R
ratio with some nonradioactive carbon-12. When a living organism dies, the carbon-14
begins to decay at a fixed rate. The formula P(t) = 100e −0.000121t gives the percentage
of carbon-14 present at time t years. Notice that when t = 0, the percentage of carbon-
14 present is 100%. When the preserved bodies of 15-year-old La Doncella and two
younger children were found in Argentina in 2005, 93.5% of the carbon-14 remained in
Extra support when you need it. their bodies, indicating that the three had died about 550 years earlier.
EXAMPLES marked with the Calc Clip (a) What is the rate of change of the percentage of carbon-14 present
in a 550-year-old fossil?
button are supported by short Natacha Pisarenko/AP Images (b) What is the rate of change of the percentage of carbon-14 present
in a 2000-year-old fossil?
CALC CLIP
video clips in that walk you Solution
through each step in the process of The perfectly preserved mummy of La (a) The rate of change of P is given by its derivative
Doncella, a 15-year-old girl, is displayed in a
solving a similar problem. museum in Salta, Argentina. d −0.000121t −0.000121t −0.000121t
�
P (t) = 100e = 100 −0.000121e =−0.0121e
dt ↑
d e u(x) = e u(x) du
dx dx
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