1 / 17

# Derivatives of exponential and logarithmic functions - PowerPoint PPT Presentation

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!!!.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Derivatives of exponential and logarithmic functions' - menefer

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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:

Definition of the

derivative!!!

The limit we just figured!

• The derivative of this

function is itself!!!

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

of logarithms to write in terms of :

Substitution!

Imp. Diff.

First off, how am I able to express in the

following way???

COB Formula!

(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…

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!!!

Find :

Find :

Find :

Find :

Find :

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

Find using logarithmic differentiation:

Differentiate:

Find using logarithmic differentiation:

Substitute:

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…

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: