Derivatives of exponential and logarithmic functions

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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:

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 COB Formula! 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 :

Use Logarithmic Differentiation: 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: