Chapter 4 Additional Derivative Topics Section 2 Derivatives of Exponential and Logarithmic Functions Objectives for Section 4.2 Derivatives of Exp/Log Functions The student will be able to calculate the derivative of ex and of ln x. The student will be able to compute the derivatives of other logarithmic and exponential functions. The student will be able to derive and use exponential and

logarithmic models. Barnett/Ziegler/Byleen Business Calculus 12e 2 The Derivative of ex eh 1 lim 1 h 0 h We will use (without proof) the fact that We now apply the four-step process from a previous section to the exponential function. Step 1: Find f (x+h) f ( x h) e x h

x e e h Step 2: Find f (x+h) f (x) x h x x h

f (x h) f (x) e e e e e 1 Barnett/Ziegler/Byleen Business Calculus 12e 3 The Derivative of ex (continued) Step 3: Find f ( x h) f ( x ) h f ( x h) f ( x) e x e h 1 h h f ( x h) f ( x ) Step 4: Find lim h 0 h h f ( x h) f ( x )

e 1 x lim e lim e x h 0 h 0 h h Barnett/Ziegler/Byleen Business Calculus 12e 4 The Derivative of ex (continued) Result: The derivative of f (x) = ex is f (x) = ex. Caution: The derivative of ex is not x ex1 The power rule cannot be used to differentiate the exponential function. The power rule applies to exponential

forms xn, where the exponent is a constant and the base is a variable. In the exponential form ex, the base is a constant and the exponent is a variable. Barnett/Ziegler/Byleen Business Calculus 12e 5 Examples Find derivatives for f (x) = ex/2 f (x) = ex/2 f (x) = 2ex + x2 f (x) = 7xe 2ex + e2 Barnett/Ziegler/Byleen Business Calculus 12e 6 Examples (continued) Find derivatives for f (x) = ex/2

f (x) = ex/2 f (x) = ex/2 f (x) = (1/2) ex/2 f (x) = 2ex +x2 f (x) = 2ex + 2x f (x) = 7xe 2ex + e2 f (x) = 7exe-1 2ex Remember that e is a real number, so the power rule is used to find the derivative of xe. The derivative of the exponential function ex, on the other hand, is ex. Note also that e2 7.389 is a constant, so its derivative is 0. Barnett/Ziegler/Byleen Business Calculus 12e 7

The Natural Logarithm Function ln x We summarize important facts about logarithmic functions from a previous section: Recall that the inverse of an exponential function is called a logarithmic function. For b > 0 and b 1 Logarithmic form is equivalent to Exponential form y = logb x x = by Domain (0, ) Domain ( , ) Range ( , )

Range (0, ) The base we will be using is e. ln x = loge x Barnett/Ziegler/Byleen Business Calculus 12e 8 The Derivative of ln x We are now ready to use the definition of derivative and the four step process to find a formula for the derivative of ln x. Later we will extend this formula to include logb x for any base b. Let f (x) = ln x, x > 0. Step 1: Find f (x+h) f ( x h) ln( x h) Step 2: Find f (x + h) f (x) xh f ( x h) f ( x) ln( x h) ln( x) ln h Barnett/Ziegler/Byleen Business Calculus 12e

9 The Derivative of ln x (continued) Step 3: Find f ( x h) f ( x ) h f ( x h) f ( x ) 1 x h 1 x h 1 h ln ln 1 ln 1 h h x x h x x x Step 4: Find x/h

f ( x h) f ( x ) lim . Let s = x/h. h 0 h f ( x h) f ( x ) 1 1 1 1/ s lim lim ln 1 s ln e h 0 h x s 0 x x Barnett/Ziegler/Byleen Business Calculus 12e 10 Examples

Find derivatives for f (x) = 5 ln x f (x) = x2 + 3 ln x f (x) = 10 ln x f (x) = x4 ln x4 Barnett/Ziegler/Byleen Business Calculus 12e 11 Examples (continued) Find derivatives for f (x) = 5 ln x f (x) = 5/x f (x) = x2 + 3 ln x f (x) = 2x + 3/x f (x) = 10 ln x

f (x) = 1/x f (x) = x4 ln x4 f (x) = 4 x3 4/x Before taking the last derivative, we rewrite f (x) using a property of logarithms: ln x4 = 4 ln x Barnett/Ziegler/Byleen Business Calculus 12e 12 Other Logarithmic and Exponential Functions Logarithmic and exponential functions with bases other than e may also be differentiated. d 1 1 log b x

dx ln b x d x x b b ln b dx Barnett/Ziegler/Byleen Business Calculus 12e 13 Examples Find derivatives for f (x) = log5 x f (x) = 2x 3x f (x) = log5 x4 Barnett/Ziegler/Byleen Business Calculus 12e 14

Examples (continued) Find derivatives for f (x) = f (x) = log5 x 1 1 ln 5 x f (x) = 2x ln 2 3x ln 3 f (x) = 2x 3x f (x) = log5 x 4 4 1 f (x) = ln 5 x For the last example, use log5 x4 = 4 log5 x

Barnett/Ziegler/Byleen Business Calculus 12e 15 Summary Exponential Rule d x x e e dx Log Rule d 1 ln x dx x For b > 0, b 1

d x b b x ln b dx d 1 1 log b x ( ) dx ln b x Barnett/Ziegler/Byleen Business Calculus 12e 16 Application On a national tour of a rock band, the demand for T-shirts is given by p(x) = 10(0.9608)x where x is the number of T-shirts (in thousands) that can be sold during a single concert at a price of $p. 1. Find the production level that produces

the maximum revenue, and the maximum revenue. Barnett/Ziegler/Byleen Business Calculus 12e 17 Application (continued) On a national tour of a rock band, the demand for T-shirts is given by p(x) = 10(0.9608)x where x is the number of T-shirts (in thousands) that can be sold during a single concert at a price of $p. 1. Find the production level that produces the maximum revenue, and the maximum revenue. R(x) = xp(x) = 10x(0.9608)x Graph on calculator and find maximum. Barnett/Ziegler/Byleen Business Calculus 12e 18

Application (continued) 2. Find the rate of change of price with respect to demand when demand is 25,000. Barnett/Ziegler/Byleen Business Calculus 12e 19 Application (continued) 2. Find the rate of change of price with respect to demand when demand is 25,000. p(x) = 10(0.9608)x(ln(0.9608)) = 0.39989(0.9608)x Substituting x = 25: p(25) = -0.39989(0.9608)25 = 0.147. This means that when demand is 25,000 shirts, in order to sell an additional 1,000 shirts the price needs to drop 15 cents. (Remember that p is measured in thousands of shirts). Barnett/Ziegler/Byleen Business Calculus 12e

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