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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: