4.3 Differentiation of Logarithmic and Exponential Functions

Graph: y1 = ln x

y2 = nDeriv(y1,x,x)

Explore the graph of y2 which is the “numeric” derivative of ln(x)

Derivative of ln x for x > 0

Examples

Consider the composite function

Chain Rule for Logarithmic Functions

If u(x) is differentiable function of x, then

Examples

Graph: y1 = e ^ x

y2 = nDeriv(y1,x,x)

Explore the graph of y2 which is the “numeric” derivative of

Derivative of Exponential Function

Examples

Consider the composite function

Chain Rule for Exponential Functions

If u(x) is a differentiable function of x, then

Examples

Find the largest and smallest value (absolute max and min) over the prescribed closed interval.

#40

Application

#74 COMPOUND INTEREST

Money is deposited in a bank offering interest at an annual rate of 6% compounded continuously. Find the percentage rate of change of the balance with respect to time.

#80 ALCOHOL ABUSE CONTROL

Suppose the percentage of alcohol in the blood t hours after consumption is given by .

a) At what rate is the blood alcohol level changing at time t?

b) How much time passes before the blood alcohol level begins to decrease?

c) Suppose the legal limit for blood alcohol is 0.04%. How much time must pass before the blood alcohol reaches this level? At what rate is the blood alcohol level decreasing when it reaches the legal limit (the 2^nd time)?

Assignment: Page 335, 1-31 odd, 39, 45, 47, 51, 73