3.9: Derivatives of Exponential and Logarithmic Functions.

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3.9 Derivatives of Exponential and Logarithmic Functions

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Presentation transcript:

3.9: Derivatives of Exponential and Logarithmic Functions

Look at the graph of The slope at x=0 appears to be 1. If we assume this to be true, then: definition of derivative

Now we attempt to find a general formula for the derivative of using the definition. This is the slope at x=0, which we have assumed to be 1.

is its own derivative! If we incorporate the chain rule: We can now use this formula to find the derivative of

( and are inverse functions.) (chain rule)

( is a constant.) Incorporating the chain rule:

Example At what point on the graph of the function y = 2 t – 3 does the tangent line have slope 21?

Example At what point on the graph of the function y = 2 t – 3 does the tangent line have slope 21?

So far today we have: Now it is relatively easy to find the derivative of.

To find the derivative of a common log function, you could just use the change of base rule for logs: The formula for the derivative of a log of any base other than e is:

Example A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the value of m?

Example A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the value of m?

Example Find dy/dx if y = log a a sin x.

Example Find dy/dx if y = log a a sin x.



Example