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

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Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x 199
NEED TO REVIEW? The natural logarithm In Chapter 1, we found that the natural logarithm function y = ln x is continuous
function is defined in Section P.5, p. 47. on its domain {x|x > 0}. Below we give the rule for finding the derivative of y = ln x.

THEOREM Derivative of the Natural Logarithm Function y = ln x
The derivative of the natural logarithm function y = ln x, x > 0, is

Skill BuildingBFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
d 1
′
y = ln x = (2)
dx x
We do not have the necessary mathematics to prove (2) now. We will prove the
theorem in Chapter 3.

EXAMPLE 8 Differentiating a Function Involving y = ln x
2
Find the derivative f (x) = 3 ln x − 5x .
Solution
2
The function f is the difference between 3 ln x and 5x . Then using (2), we find that
d d d d 3
2
′ 2 (5x ) = 3
f (x) = (3 ln x − 5x ) = (3 ln x) − (ln x) − 5 · 2x = − 10x
dx ↑ dx dx ↑ dx ↑ x
Difference Rule Constant Multiple Rule; Use (2).
Simple Power Rule
R
NOW WORK Problem 23 and AP Practice Problems 7 and 9.

2.3 Assess Your Understanding

Concepts and Vocabulary
d d
PAGE 2 3 3 7
194 1. π = ; x = . t + 2 x − 5x
dx dx 17. f (t) = 18. f (x) =
5 9
2. When n is a positive integer, the Simple Power Rule 3
x + 2x + 1 1 2
d 19. f (x) = 20. f (x) = (ax + bx + c), a 6= 0
n
states that x = . 7 a
dx 1
3. True or False The derivative of a power function of degree 21. f (x) = 4e x 22. f (x) = − e x
2
greater than 1 is also a power function.
PAGE
199 23. f (x) = x − ln x 24. f (x) = 5 ln x + 8
4. If k is a constant and f is a differentiable function,
PAGE u u
d 198 25. f (u) = 5 ln u − 2e 26. f (u) = 3e + 10 ln u
then [k f (x)] = .
dx
x
5. The derivative of f (x) = e is . In Problems 27–32, find each derivative.
4
d √ 1 d 2t − 5
6. True or False The derivative of an exponential 27. 3 t + 28.
x
function f (x) = a , where a > 0 and a 6= 1, is always dt 2 dt 8
x
a constant multiple of a . d A 2 dC
29. if A(R) = π R 30. if C = 2π R
d R 4 3 d R
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dV
d P
PAGE
3
In Problems 7–26, find the derivative of each function using the 195 31. dr if V = πr 32. dT if P = 0.2T
formulas of this section. (a, b, c, and d, when they appear, are In Problems 33–36:
constants.) (a) Find the slope of the tangent line to the graph of each
PAGE √ function f at the indicated point.
195 7. f (x) = 3x + 2 8. f (x) = 5x − π (b) Find an equation of the tangent line at the point.
4
2
2
9. f (x) = x + 3x + 4 10. f (x) = 4x + 2x − 2
(c) Find an equation of the normal line at the point.
5
3
2
11. f (u) = 8u − 5u + 1 12. f (u) = 9u − 2u + 4u + 4 (d) Graph f and the tangent line and normal line found
3 in (b) and (c) on the same set of axes.
3
13. f (s) = as + s 2 14. f (s) = 4 − πs 2
2 PAGE
4
3
1 1 197 33. f (x) = x + 3x − 1 at (0, −1) 34. f (x) = x + 2x − 1 at (1, 2)
6
2
8
15. f (t) = (t − 5t) 16. f (x) = (x − 5x + 2)
6 8 35. f (x) = e + 5x at (0, 1) 36. f (x) = 4 − e at (0, 3)
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