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Review: properties of ln 1) 2) 3) 4) 5)

Let’s see if we can discover why the rule is as above. First define the natural log function as follows: Now rewrite in exponential form: Now differentiate implicitly:

Example 6: Find the derivative of   Solution: This derivative will require the product rule.     Product Rule: (1st)(derivative of 2nd) + (2nd)(derivative of 1st)  

Solution: This derivative will require the quotient rule.   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: (1st)(derivative of 2nd) + (2nd)(derivative of 1st) Try again. Return to previous slide.

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

Example 8: Find the derivative of Solution: Using the chain rule for logarithmic functions. Derivative of the inside, x²+1 The inside, x²+1

Example 9: Differentiate Solution: There are two ways to do this problem.

The other 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. You’ll notice this is the same as the first solution.

Example 10: Differentiate Solution: Using what we learned in the previous example. Expand first: Now differentiate: Recall lnex = x

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 11: 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 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…

We learned two rules for differentiating logarithmic functions: Rule 2: Derivative of ln x Rule 3: 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.