5.1 The Natural Logarithmic Function: Differentiation

After this lesson, you should be able to: Develop and use properties of the natural logarithmic function. Understand the definition of the number e. Find derivatives of functions involving the natural logarithmic function.

Now we will find derivatives of logarithmic functions and we will need rules for finding their derivatives. Derivative of ln x Let ’ s see if we can discover why the rule is as above. First define the natural log function as follows: Now differentiate implicitly: Now rewrite in exponential form:

Theorem 5.1 Properties of the Natural Logarithmic Function

Example 1 Find the derivative of f(x)= xlnx. Solution This derivative will require the product rule. Product Rule: (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st )

Example 2 Find the derivative of g(x)= lnx/x Solution This derivative will require the quotient rule. Quotient Rule: (bottom)(derivative of top) – (top)(derivative of bottom) (bottom)²

Why don ’ t you try one: Find the derivative of y = x²lnx. The derivative will require you to use the product rule. Which of the following is the correct? y ’ = 2 y ’ = 2xlnx y ’ = x + 2xlnx

No, sorry that is not the correct answer. Keep in mind - Product Rule: (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st ) Try again.

F ’ (x) = (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st ) Good work! Using the product rule: y ’ = x² + (lnx)(2x) y ’ = x + 2xlnx This can also be written y ’ = x(1+2lnx)

Theorem 5.2 Logarithmic Properties

The Chain Rule for Log Functions Here is the second rule for differentiating logarithmic functions. In words, the derivative of the natural log of f ( x ) is 1 over f ( x ) times the derivative of f ( x ) Or, the derivative of the natural log of f(x) is the derivative of f ( x ) over f ( x )

Theorem 5.3 Derivative of the Natural Logarithmic Function

Example 3 Find the derivative of Solution Using the chain rule for logarithmic functions. Derivative of the inside, x²+1 The inside, x²+1

Example 4 Differentiate Solution There are two ways to do this problem. One is easy and the other is more difficult. The difficult way:

The easy way requires that we simplify the log using some of the expansion properties. Now using the simplified version of y we find y’. Now that you have a common denominator, combine into a single fraction.

Now we can simplify You ’ ll notice this is the same as the first solution.

Example 5 Differentiate Solution Using what we learned in the previous example. Expand first: Now differentiate: Recall ln e x = x

Now you try: Find the derivative of. Following the method of the previous two examples. What is the next step?

This method of differentiating is valid, but it is the more difficult way to find the derivative. It would be simpler to expand first using properties of logs and then find the derivative. Click and you will see the correct expansion followed by the derivative.

Correct. First you should expand to Then find the derivative using the rule 4 on each logarithm. Now get a common denominator and simplify.

Example 6 Differentiate Solution Although this problem could be easily done by multiplying the expression out, I would like to introduce to you a technique which you can use when the expression is a lot more complicated. Step 1 Take the natural log ln of both sides. Step 2 Expand the complicated side. Step 3 Differentiate both side (implicitly for ln y )

Step 4: Solve for y’. Step 5: Substitute y in the above equation and simplify.

Continue to simplify…

Let ’ s double check to make sure that derivative is correct by Multiplying out the original and then taking the derivative. Remember this problem was to practice the technique. You would not use it on something this simple.

Example 7 Consider the function y = x x. Not a power function nor an exponential function. This is the graph: domain x > 0 What is that minimum point? Recall to find a minimum, we need to find the first derivative, find the critical numbers and use either the First Derivative Test or the Second Derivative Test to determine the extrema.

To find the derivative of y = x x, we will take the ln of both sides first and then expand. Now, to find the derivative we differentiate both sides implicitly.

To find the critical numbers, set y’ = 0 and solve for x. Thus, the minimum point occurs at x = 1/ e or about Now test x = 0.1 in y’, y’ (0.1) = < 0 and x = 0.5 in y’, y’ (0.5) = > 0

Summary We learned two rules for differentiating logarithmic functions: Rule 1: Derivative of lnx Rule 2: The Chain Rule for Log Functions We also learned it can be beneficial to expand a logarithm before you take the derivative and that sometimes it is useful to take the natural log ( ln ) of both sides of an equation, rewrite and then take the derivative implicitly.

Homework Section 5.1 Pg odds, odds, 71, 75, 77, 79, 83, 85