Section 3.7 Derivatives of logarithmic functions 1 Rules of - PowerPoint PPT Presentation

Section 3.7 “Derivatives of logarithmic functions” 1

Rules of exponentials and logarithms 1. a b + c = a b a c 1. log a ( bc ) = log a ( b ) + log a ( c ) 2. a b − c = a b 2. log a b c = log a ( b ) − log a ( c ) a c 3. log a ( b c ) = c log a b 3. ( a b ) c = a bc a log a b = b 4. log a 1 4. a − 1 = 1 b = − log a b a 5. a 0 = 1. 5. log a 1 = 0. 2

Derivatives of logarithmic functions Theorem. 1. dx ln x = 1 d x. 2. If a > 0 , d 1 dx log a x = (ln a ) x. 3

Derivatives of exponential functions Theorem. If a is any positive number, d dxa x = (ln a ) a x . 4

Tips for Logarithmic Differentiation 1. Start with y = f ( x ). 2. Take the natural logarithm of both sides. 3. Use properties of logarithms to simplify the right-hand side. 4. Take the derivative. On the left you will have dx ln y = 1 d dy dx. y 5. Multiply both sides by y and substitute y = f ( x ). 5

Theorem. h → 0 ln (1 + h ) 1 /h = 1 . lim Proof. Let f ( x ) = ln x . We know f ′ (1) = 0. This means f (1 + h ) − f (1) 1 = lim h h → 0 ln(1 + h ) − ln 1 = lim h h → 0 h → 0 ln(1 + h ) 1 /h . = lim 6

Theorem. � n 1 + 1 � lim = e. n →∞ n Proof. First, exponentiate the last theorem: e lim h → 0 ln(1+ h ) 1 /h = e 1 h → 0 e ln(1+ h ) 1 /h = e 1 lim h → 0 (1 + h ) 1 /h = e. lim Now, if n is any positive number, let h = 1 n . Then as n → ∞ , h → 0, and so � n 1 + 1 � h → 0 (1 + h ) 1 /h = e. lim = lim n →∞ n 7

Questions 8

dx ln( π ) = 1 d 1. [Q] True or False . π . 9

� n 1 + 1 � 2. [Q] Your calculus book says that e = lim . This means: n →∞ n (a) e is not really a number because it is a limit (b) e cannot be computed � 2 � 3 � 100 � 2 � 3 � 4 � 101 � (c) the sequence of numbers , , , ..., , ... get 1 2 3 100 as close as you want to the number e 10

3. [P] When you read in the newspaper thing like inflation rate, interest rate, birth rate, etc., it always means f ′ f , not f ′ itself. True or False . f ′ f is not the derivative of a function. 11