The document discusses the derivatives of exponential functions. It calculates the derivative of f(x)=2^x and finds that the derivative of 2^x is proportional to 2^x with a constant of proportionality of 0.693. It also shows values of the derivative for different values of a in f(x)=a^x and determines that the graph of these values follows the function y=ln(a).

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

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

3. 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

4. The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. .69556 0.01 .69339 0.001 .69317 0.0001 .69312 -0.0001 .69291 -0.001 .69075 -0.01 h

5. 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 .69556 0.01 .69339 0.001 .69317 0.0001 .69312 -0.0001 .69291 -0.001 .69075 -0.01 h

6. 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

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

8. The Derivatives of Exponential Functions Calculate the derivative of f ( x ) =a x

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

10. Here is for different values of a 1.946 1.797 1.609 1.386 1.0986 0.693 7 6 5 4 3 2 a

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

12. Here is for different values of a 1.946 1.797 1.609 1.386 1.0986 0.693 7 6 5 4 3 2 a Turns out that the graph is just y = ln( a )

13. Here is for different values of a ln(7) = 1.946 ln(6) = 1.797 ln(5) = 1.609 ln(4) = 1.386 ln(3) = 1.0986 ln(2) = 0.693 7 6 5 4 3 2 a