HERE IS SOME BRIEF HELP FROM THE BOOK YOU CAN USE TO STUDY FOR NEXT CLASS! THIS IS REVIEW FROM PAGES 234-237. I HOPE THIS HELPS SOME!

First off, what is an exponential function?

A exponential function is a function whose value is a constant raised to the power of the argument.

For example, f(x)=, where a > 0 and a 1

FINDING THE DERIVATIVE:

We can find the derivative of exponential functions by using the definition of the derivative.

= (assuming a > 0)

=

=

In this last step, since does not involve h, we were able to bring in front of the limit. The result says that the derivative of is times a constant that depends on a, namely .

We could also find the derivative with the following short cuts:

derivative of

This formula becomes particularly simple when we let a = e, because of the face that ln e = 1.

Derivative of

We now see why e is the best base to work with: it has the simplest derivative of all exponential functions. Even if we choose a different base, e appears in the derivative anyway through the ln a term. (Recall that ln a is tha logarithm of a to the base e.) In fact, of all the functions we have studied, is the simplest to differentiate, because its derivative is just itself.

Derivative of and

and

EXAMPLES:

**(A)** y=

Solution: let g(x) = 5x, with g'(x) =5 then

**(B) **

Solution: use the product rule

+

=

=

**(C) **f(x)=

Solution: use the quotient rule.

f(x) =

=

******Here is a joke that is actually in our textbook:*****

A deranged mathematician who frightened other inmates at an insane asylum by screaming at them, “I am going to differentiate you!” But one inmate remained calm and simply responded, ” I don’t care; I’m .”