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Derivatives of exponential and logarithmic functionsPowerPoint Presentation

Derivatives of exponential and logarithmic functions

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

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Presentation Transcript

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 :

(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!!!

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 :

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:

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