The document discusses calculating the derivative of exponential functions like f(x)=2^x. It provides a table of values of (2^x - 2^(x+h))/h as h approaches 0, showing the limit is approximately 0.693. This suggests the derivative of 2^x is proportional to 2^x with a constant of 0.693. A second table shows how the derivative of a^x is related to the value of a, with the derivative being a when a=1.

1. The Derivatives of Exponential Functions Calculate the derivative of f ( x ) =2 x What is this????

2. The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. 0.01 0.001 0.0001 -0.0001 -0.001 -0.01 h

3. The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. This table suggests that the limit DOES exist, and has a value of about 0.693 So we can write: .69556 0.01 .69339 0.001 .69317 0.0001 .69312 -0.0001 .69291 -0.001 .69075 -0.01 h

4. So the derivative of 2 x is proportional to 2 x with a constant of proportionality 0.693. Hmmmm…

5. The Derivatives of Exponential Functions Calculate the derivative of f ( x ) =a x What is this????

6. Here is for different values of a 2.079 1.946 1.797 1.609 1.386 1.0986 0.693 7 8 6 5 4 3 2 a Use your calculator to plot these points. What type of function does it look like?