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1、5.5 Differentiation of Logarithmic FunctionsByDr. Julia Arnold and Ms. Karen Overmanusing Tans 5th edition Applied Calculus for the managerial , life, and social sciences textNow we will find derivatives of logarithmic functions and we will Need rules for finding their derivatives.Rule 3: Derivative
2、 of ln xLets 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:Example 1: Find the derivative of f(x)= xlnx.Solution: This derivative will require the product rule. Product Rule:(1st)(derivat
3、ive of 2nd) + (2nd)(derivative of 1st)Example 2: Find the derivative of g(x)= lnx/xSolution: This derivative will require the quotient rule.Quotient Rule:(bottom)(derivative of top) (top)(derivative of bottom) (bottom)Why dont you try one: Find the derivative of y = xlnx . The derivative will requir
4、e you to use the product rule.Which of the following is the correct?y = 2y = 2xlnxy = x + 2xlnxNo, 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.F(x) = (1st)(derivative of 2nd) + (2nd)(derivati
5、ve of 1st)Good work! Using the product rule:y = x + (lnx)(2x)y = x + 2xlnxThis can also be written y = x(1+2lnx)Rule 4: The Chain Rule for Log FunctionsHere 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 deriv
6、ative of f(x) Or, the derivative of the natural log of f(x) is the derivative of f(x) over f(x)Example 3: Find the derivative of Solution: Using the chain rule for logarithmic functions.Derivative of the inside, x+1The inside, x+1 Example 4: Differentiate Solution: There are two ways to do this prob
7、lem. 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 singlefraction.Youll notice this is the sa
8、me as the first solution.Example 6: DifferentiateSolution: Using what we learned in the previous example.Expand first:Now differentiate:Recall lnex=xFind 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
9、more difficultway to find the derivative. It would be simplier 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
10、.Now get a common denominator and simplify.Example 7: 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.S
11、tep 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 simplifyLets double check to make sure that derivative is correct byMultiplying out the original and then taking the derivati
12、ve.Remember this problem was to practice the technique. You would not use it on something this simple.Consider the function y = xx. Not a power function nor an exponential function.This is the graph: domain x 0What is that minimum point?Recall to find a minimum, we need to find the first derivative,
13、 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 = xx , 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 critica
14、l numbers, set y = 0 and solve for x.Thus, the minimum point occurs at x = 1/e or about .37Now test x = 0.1 in y, y(0.1) = -1.034 0We learned two rules for differentiating logarithmic functions:Rule 3: Derivative of ln xRule 4: The Chain Rule for Log FunctionsWe also learned it can be beneficial to
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