Learning Goals: 3.9: Derivatives of Exponential & Logarithmic Functions Derivatives of exponential and logarithmic functions. Logarithmic differentiation.

Properties of e The definition: Proof is tuff! Support follows

Important Idea The derivative of the natural exponential function Chain Rule Version e x is its own derivative

Proof

Example Find the derivative:

Try This Find the derivative: Did you use the Product Rule?

Try This Find the derivative:

Try This Find the derivative:

Try This The first permanent colony in America was established in Jamestown, Virginia, in For 1610 through 1780, the population, p ( in thousands), in colonial America can be modeled by... where t =10 represents the year1610. At what rate was the population of colonial America changing in 1650? Include the correct units. The population is increasing by 2369 people per year.

Natural Logarithm y = e x x = e y ln(x) = ln(e y ) ln(x) = yln(e) y = ln(x) x = e y

Therefore The chain rule version:

Examples

Try This

Are you using the product rule?

Try This Are you using the quotient rule?

Example

Is this a quotient rule problem? Maybe!?

In previous examples, we used: But, what about... Important Idea

Chain Rule Version

Compare: with: Note: ln a is a constant factor Why doesn’t the other have a constant factor? Important Idea

Example Find the derivative: Hint: This is not a power rule question!

Try This Find the derivative:

Example or rewrite

Try This Find the derivative: Hint: Rewrite using the log properties and then use the chain rule

Solution Rewrite:

Solution Use chain rule:

Try This Rewrite using log properties before differentiation...

Rewrite: Solution

…then differentiate Solution And simplify:

Definition Since ln x is not defined for negative values of x, you may frequently see ln|x|. The absolute value rule for ln is: When differentiating a logarithm, you may ignore any absolute value sign.

Try This Find the derivative: Don’t forget the chain rule

Try This Find the equation of the line tangent to: at (1,1)

Example If,find and state the domain of and

BOOKMARK

Important Idea Chain Rule Version

Compare with

Try This Find the derivative:

Try This Find the derivative: Hint:

Examples Find the derivative: HOW?

OMG!

Let’s try something else…

Example Using Logarithmic Differentiation: Find for Steps: 1. Take ln of both sides. 3. Differentiate implicitly. 2. Use properties of logs.

Example Logarithmic Differentiation: Find for

Try This Use logarithmic differentiation to find :

Write with a common denominator.

Lesson Close The derivatives of exponential and logarithmic functions are important in the mathematics of engineering, science and business. The mathematics of exponential and logarithmic functions will be extensively tested on the AP Exam.