Page 78 - 2024-calc4e-SE proofs-4e.indd

Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

228 Chapter 2 • The Derivative and Its Properties

• Theorem If a function f has a derivative at a number c, d d
then f is continuous at c. (p. 184) f g − f g
d f dx dx
• Corollary If a function f is discontinuous at a number c, • Quotient Rule dx g = g 2
then f has no derivative at c. (p. 185) (p. 206)
′
′
f f g − f g ′
2.3 The Derivative of a Polynomial Function; g = g 2
x
The Derivative of y = e and y = ln x
provided g(x) 6= 0
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
dy d d
• Leibniz notation = y = f (x) (p. 192) d g
dx dx dx d 1
• Reciprocal Rule = − dx
• Basic derivatives dx g g 2
(p. 207)
d d ′ g ′
1
A = 0 A is a constant (p. 192) x = 1 (p. 193)
dx dx g = − g 2
d d 1
x
x
e = e (p. 198) ln x = (p. 199) provided g(x) 6= 0
dx dx x
d
n
• Simple Power Rule x = nx n − 1 , n an integer (p. 208)
d dx
n
• Simple Power Rule x = nx n − 1 , n ≥ 1, an integer
dx • Higher-order derivatives See Table 3 (p. 209)
(p. 193)
• Position Function s = s(t) (p. 210)
Properties of Derivatives
d d d ds
• Sum Rule [ f + g] = f + g • Velocity v = v(t) = (p. 210)
(p. 195) dx dx dx dt
2
′
( f + g) = f + g ′ dv d s
′
• Acceleration a = a(t) = = (p. 210)
dt dt 2
d d d
• Difference Rule [ f − g] = f − g
(p. 196) dx dx dx 2.5 The Derivative of the Trigonometric Functions
′
′
( f − g) = f − g ′
Basic Derivatives
• Constant Multiple Rule (p. 194) If k is a constant,
d d
sin x = cos x (p. 218) sec x = sec x tan x (p. 221)
d d dx dx
[k f ] = k f
dx dx d d
′
(k f ) = k · f ′ cos x = −sin x (p. 219) csc x = −csc x cot x (p. 221)
dx dx
2.4 Differentiating the Product and the Quotient of d 2 d 2
Two Functions; Higher-Order Derivatives dx tan x = sec x (p. 221) dx cot x = −csc x (p. 221)
Properties of Derivatives

• Product Rule d d d
( f g) = f g + f g
(p. 204) dx dx dx
′
′
( f g) = f g + f g
′
Preparing for the
OBJECTIVES R
AP Exam
R
AP Review Problems
Section You should be able to ... Examples Review Exercises
2.1 1 Find equations for the tangent line and the normal line 1 67–70 7, 10
to the graph of a function (p. 168)
2 Find the rate of change of a function (p. 169) 2, 3 1, 2, 73 (a) 6
3 Find average velocity and instantaneous velocity (p. 170) 4, 5 71(a), (b); 72(a), (b) 12
4 Find the derivative of a function at a number (p. 173) 6–9 3–8, 75 5, 11
2.2 1 Define the derivative function (p. 179) 1–3 9–12 2, 13
2 Graph the derivative function (p. 181) 4, 5 9–12, 15–18
3 Identify where a function is not differentiable (p. 182) 6–8 13, 14, 75 4
4 Explain the relationship between differentiability 9, 10 13, 14, 75 4
and continuity (p. 184)
2.3 1 Differentiate a constant function (p. 192) 1
2 Differentiate a power function; the simple 2, 3 19–22
power rule (p. 192)
3 Differentiate the sum and the difference of two 4–6 23–26, 33, 34, 40, 51, 52, 67 6, 8, 12
functions (p. 195)
4 Differentiate the exponential function y = e x 7, 8 44, 45, 53, 54, 56, 59, 69 6, 7, 9
and the natural logarithm function y = ln x (p. 197)
© 2024 BFW Publishers PAGES NOT FINAL
For Review Purposes Only, all other uses prohibited
Do Not Copy or Post in Any Form.