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

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

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

The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. h

The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. h This table suggests that the limit DOES exist, and has a value of about So we can write:

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

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

Here is for different values of a a Use your calculator to plot these points. What type of function does it look like?

RESULTS Consequently,

EXAMPLES

The Derivative of ln x Numerically estimate the derivative at the following input values. xDerivative of ln x x

The Derivative of ln x Numerically estimate the derivative at the following input values. xDerivative of ln x x1/x

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

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