Simple Rules for Differentiation



Exponential and Logarithmic Functions

Objectives:

Students will be able to

• Calculate derivatives of exponential functions

• Calculate derivatives of logarithmic functions

So far we have looked at derivatives of power functions ([pic]) and where a is a real number and derivatives of function that are made by adding, subtracting, multiplying, and dividing power functions and functions that are made by composing power functions.

There are other functions that we have seen in the past – exponential and logarithmic functions. We need to be able to take derivatives of these types of functions as well.

Exponential Functions

The natural exponential function is a function that has a very special property (see Sydsaeter text for proof) that for the function [pic], [pic] (the derivative of the function is the function itself).

For a general exponential function [pic], [pic]. Remember here a is a positive number not equal to 1.

Logarithmic Functions

Just like with the exponential functions, we are going to start with the natural logarithmic function. For the function [pic], the derivative [pic].

For the general logarithmic function [pic], [pic]. Here again a is a positive number not equal to 1.

Example 1:

Calculate the derivative of the function [pic]

Example 2:

Calculate the derivative of the function [pic]

Example 3:

Calculate the derivative of the function [pic] with respect to z

Example 4:

Calculate the derivative of the function [pic] with respect to t

Example 5:

Calculate the derivative of the function [pic]

Example 6:

Calculate the derivative of the function [pic]

Example 7:

Calculate the derivative of the function [pic]

Example 8:

Find the second derivative of the function [pic]

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