Mat210 Section 1.11b

Mat210 Section 1.11b Derivatives of Exponential and Logarithmic Functions

How do you find the derivative of an exponential function?

In this lesson we examine methods for finding derivatives of exponential and logarithmic functions. In particular, we provide some "coaching tips" how to find the derivative of four function types:

(1) exponential and logarithmic with base "e" (case 1 below)

(2) exponentials and logarithms with base "e" and a chain rule component. (case 2 below)

(3) exponentials and logarithms with base "a", which is any base other than e, for example, base 10. (case 3 below)

(4) exponentials and logarithms with base "a" and a chain rule component. (case 4 below)

It is important to keep in mind that the aim here in this lesson is in no way different than in the previous sections on derivatives, which is to find a function that gives a rate of change of the original model or equation. The objective is to be able to find derivatives of functions using methods that are unlike the methods we learned for polynomials. You already know about case 1. Learn to spot which of the remaining cases you have in any given problem, and the method of finding the derivative will become simple.

Case 1
e to the power of x

Case 2
e to the power of f(x)

Case 3
any positive real number a to the power of x

Case 4
any positive real number a to the power of f(x)

Function type

same as

Derivative

(chain rule)

It is quite helpful to look for patterns and similarities while you are learning these methods.

Example 1: Find the derivatives of the following functions. Roll your mouse over the images below the problem to check your answers.

Coaching tips: note that these are all Case 1 type problems! Even though you have to apply a product rule in the last two instances, you actually only needed to know that "the derivative of e ^x is e^x".

Example 2: Find . Roll your mouse over the images below the problem to check your answers.

Coaching tips: OK, these were a little tougher, but if you look closely you will see that they are all "case 2" type problems, where the additional calculations from the chain rule come into play. The main features of the derivative in Case 2 type problems are (1) you do not reduce the exponent by 1 (2) the derivative of the function is itself (as in Case 1) times the derivative of the exponent (chain rule).

Example 3: Find . Roll your mouse over the images below the problem to check your answers.

Coaching tips Yes, you guessed it, these are all Case 3 types. How these differ from case 2 is very straightforward. You find the derivative as you did in case 2, then you multiply by the natural log of the base. In other words, the derivative lo9oks the same as with base e, except we multiply by "ln(a)" also, where "a" is the base.

Example 4: Find . Roll your mouse over the images below the problem to check your answers.

Coaching tips And finally, these are all Case 4 types. How these differ from case 3 is simple. You find the derivative as you did in case 3, but you have a chain rule component, so you have to multiply by the derivative of the exponent.

Take a breather, and when you are ready for round 2, get ready for a very similar set of patterns.

Well, dog-gone-it, ready for round 2? We now analyze how to find the derivative of a variety of logarithmic functions.

	Case 1 the natural log of x	Case 2 the natural log of f(x)	Case 3 log to any positive real base a of x	Case 4 log to any positive real base a of f of x
Function type
Derivative

Example 5: Find the derivative. Roll your mouse over the images below the problem to check your answers.

Example 6: Find Roll your mouse over the images below the problem to check your answers.

Example 7: Find Roll your mouse over the images below the problem to check your answers.

Example 8: Find Roll your mouse over the images below the problem to check your answers.