Derivatives of Exponential and Logarithmic Functions Section 3.2 Derivatives of Exponential and Logarithmic Functions

Exponential Growth What are some things you can think of that grow exponentially?

Examples of Exponential Growth and Decay Human Population growth (up to a point…)

Examples of Exponential Growth and Decay Growth of Bacteria in a Petri Dish (again, up to a point) Decay of radioactive atoms (Carbon-14) Removal of drugs from body Example: 50% of caffeine is removed from your body approximately every 4 hours

Sec 3.2 Derivatives of Exponential and Logarithmic Functions Formulas for derivatives of:

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h 𝑓 ′ 1 = lim ℎ→0 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ = lim ℎ→0 𝑓 1+0.1 −𝑓(1) 0.1

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h 𝑓 ′ 1 = lim ℎ→0 𝑓 1+0.1 −𝑓(1) 0.1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.1 − 𝑒 1 0.1 =2.859

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.1 − 𝑒 1 0.1 =2.859 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.01 − 𝑒 1 0.01 =2.732 Smaller value of h = 0.01

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.1 − 𝑒 1 0.1 =2.859 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.01 − 𝑒 1 0.01 =2.732 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.001 − 𝑒 1 0.001 =2.720 Smaller value of h = 0.001

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.1 − 𝑒 1 0.1 =2.859 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.01 − 𝑒 1 0.01 =2.732 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.001 − 𝑒 1 0.001 =2.720 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.0001 − 𝑒 1 0.0001 =2.718 Smaller value of h = 0.0001

Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h The number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are: 2.718281828459045235 (and more ...) It is often called Euler's number after Leonhard Euler.  e is the base of the Natural Logarithms (invented by John Napier). Find f’(1) for various values of h 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.1 − 𝑒 1 0.1 =2.859 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.01 − 𝑒 1 0.01 =2.732 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.001 − 𝑒 1 0.001 =2.720 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.0001 − 𝑒 1 0.0001 =2.718 𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 1+0.00001 − 𝑒 1 0.00001 =2.718 Getting close to “e”

Derivative Function of 𝑓 𝑥 = 𝑒 𝑥 Start with the limit definition of the derivative:

Derivative Function of 𝑓 𝑥 = 𝑒 𝑥 Start with the limit definition of the derivative:

Derivative Function of 𝑓 𝑥 = 𝑒 𝑥 Start with the limit definition of the derivative:

Derivative Function of 𝑓 𝑥 = 𝑒 𝑥 Start with the limit definition of the derivative:

Derivative Function of 𝑓 𝑥 = 𝑒 𝑥 Start with the limit definition of the derivative:

Try various small values of h to find limiting value -0.01 -0.001 0.001 0.01 0.995 0.9995 ? 1.0005 1.005 Try various small values of h to find limiting value

h -0.01 -0.001 0.001 0.01 0.995 0.9995 1 1.0005 1.005 Limiting Value

h -0.01 -0.001 0.001 0.01 0.995 0.9995 1 1.0005 1.005

So h -0.01 -0.001 0.001 0.01 0.995 0.9995 ? 1.0005 1.005

Example: h -0.01 -0.001 0.001 0.01 0.995 0.9995 ? 1.0005 1.005

So Example: h -0.01 -0.001 0.001 0.01 0.995 0.9995 ? 1.0005 1.005

So Example: h -0.01 -0.001 0.001 0.01 0.995 0.9995 ? 1.0005 1.005

Derivative Function for 𝑓 𝑥 = 𝑎 𝑥

Derivative Function for 𝑓 𝑥 = 𝑎 𝑥 Need the “chain rule” (coming up) to understand and prove (later)

Derivative Function for 𝑓 𝑥 = 𝑎 𝑥 Examples:

Derivative Function for 𝑓 𝑥 = 𝑎 𝑥 Examples:

Derivative Function for 𝑓 𝑥 = 𝑎 𝑥 Examples:

Derivative Function for 𝑓 𝑥 = 𝑎 𝑥 Examples:

Derivative Function for 𝑓 𝑡 = 𝑒 𝑘𝑡

Derivative Function for 𝑓 𝑡 = 𝑒 𝑘𝑡 Need the “chain rule” (coming up) to understand and prove (later)

Derivative Function for 𝑓 𝑡 = 𝑒 𝑘𝑡

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥) Will also prove after chain rule

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥) Will also prove after chain rule

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥) Will also prove after chain rule

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥) Will also prove after chain rule

Derivative Function for 𝑓 𝑥 =ln⁡(𝑥) Will also prove after chain rule

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000.

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009?

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒 = 𝑑𝑃 𝑑𝑡

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9 𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 2020 1.036 𝑡

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9 𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 2020 1.036 𝑡 =2020 𝑑 𝑑𝑡 1.036 𝑡

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9 𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 2020 1.036 𝑡 =2020 𝑑 𝑑𝑡 1.036 𝑡 =2020 ln⁡(1.036) 1.036 𝑡

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9 𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 2020 1.036 𝑡 =2020 𝑑 𝑑𝑡 1.036 𝑡 =2020 ln⁡(1.036) 1.036 𝑡

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Instantaneous Rate of Change at t = 9 𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 2020 1.036 𝑡 =2020 𝑑 𝑑𝑡 1.036 𝑡 =2020 ln 1.036 1.036 9 =98.22

Applications Example 1 Population of Nevada, P, in millions can be approximated by 𝑃=2020 1.036 𝑡 Where t is years since the start of 2000. At what rate was the population growing at the beginning of 2009? Solution 98.22 million people/year 𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 2020 1.036 𝑡 =2020 𝑑 𝑑𝑡 1.036 𝑡 =2020 ln 1.036 1.036 9 =98.22

Applications Example 2 At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter). What is the concentration 4 hours after it was administered? At what rate is the concentration changing at that time?

Applications Example 2 At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter). What is the concentration 4 hours after it was administered? 𝑓 4 =27 𝑒 −0.14(4) =15.42 𝑛𝑔/𝑚𝑙

Applications Example 2 At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter). At what rate is the concentration changing at that time? 𝑓′ 𝑡 =27 (−0.14)𝑒 −0.14𝑡

Applications Example 2 At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter). At what rate is the concentration changing at that time? 𝑓 ′ (4)=27 −0.14 𝑒 −0.14 4 =−2.16 𝑛𝑔 𝑚𝑙 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟