1. MAT 213 Brief Calculus Section 3.3 Exponential and Logarithmic Rate-of- Change Formulas

2. Below is a graph of f(x)=2 x What do you think the graph of f’(x) would look like?

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

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

5. The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. h -0.01.69075 -0.001.69291 -0.0001.69312 0.0001.69317 0.001.69339 0.01.69556 This table suggests that the limit DOES exist, and has a value of about 0.693 So we can write:

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

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

8. Here is for different values of a a 20.693 31.0986 41.386 51.609 61.797 71.946 82.079 Use your calculator to plot these points. What type of function does it look like?

9. RESULTS Consequently,

10. EXAMPLES

11. The Derivative of ln x Numerically estimate the derivative at the following input values. xDerivative of ln x 10 4 2 1 0.5 0.25 0.1 x

12. The Derivative of ln x Numerically estimate the derivative at the following input values. xDerivative of ln x 10.1 4.25 2.5 11 0.52 0.254 0.110 x1/x

13. The Derivative of ln x If y = lnx, then for x > 0. Examples

14. In groups let’s try the following from the book 23