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Derivatives of exponential and logarithmic functions

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Derivatives of exponential and logarithmic functions

Section 3.9

If you recall, the number e is important in many

instances of exponential growth:

Find the following important limit using graphs

and/or tables:

Derivative of

Definition of the

derivative!!!

The limit we just figured!

- The derivative of this
function is itself!!!

Derivative of

Given a positive base that is not one, we can use a property

of logarithms to write in terms of :

Derivative of

Substitution!

Imp. Diff.

Derivative of

First off, how am I able to express in the

following way???

COB Formula!

Summary of the New Rules

(keeping in mind the Chain Rule and any variable restrictions)

Now we can realize the FULL POWER

of the Power Rule……………observe:

Start by writing x with any real power as a power of e…

Power Rule for Arbitrary Real Powers

If u is a positive differentiable function of x and n is

any real number, then is a differentiable function

of x, and

- The power rule works for not only integers, not only rational numbers, but any real numbers!!!

Quality Practice Problems

Find :

Find :

Find :

Quality Practice Problems

Find :

Find :

Quality Practice Problems

How do we differentiate a function when both the base and exponent contain the variable???

Find :

Use Logarithmic Differentiation:

1. Take the natural logarithm of both sides of the

equation

2. Use the properties of logarithms to simplify the

equation

3. Differentiate (sometimes implicitly!) the

simplified equation

Quality Practice Problems

Find :

Quality Practice Problems

Find using logarithmic differentiation:

Differentiate:

Quality Practice Problems

Find using logarithmic differentiation:

Substitute:

Quality Practice Problems

A line with slope m passes through the origin and is tangent

to the graph of . What is the value of m?

What does the graph look like?

The slope of the curve:

The slope of the line:

Now, let’s set them equal…

Quality Practice Problems

A line with slope m passes through the origin and is tangent

to the graph of . What is the value of m?

What does the graph look like?

So, our slope: