Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log 3 2 log b M r = r log b M Ex: log = 3 log 7 10

5-1 The Natural Logarithmic Function 5-4 Exponential Functions 5-5 Bases other than e and Applications 5-3 Inverse Functions and their Derivatives 5-8 Inverse Trig Functions 7-7 Indeterminate Forms and L’Hopital’s Rule Chapter 5 Derivatives of Transcendental Functions

Goals: Find derivatives of functions involving the natural logarithmic function. Use logarithmic differentiation to take derivatives of more complicated functions. Sect. 5-1 Derivatives of Natural Logarithmic Functions

Derivatives of Natural Logarithmic Function

Example 1: Find the derivative of f(x)= xlnx. Solution: This derivative will require the product rule.

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

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

YOU TRY… 1. Find the derivative of y = x²lnx.

y’ = x² + (lnx)(2x) = x + 2xlnx = x(1+2lnx) Example 1: Solution

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: First simplify the log using some of the expansion properties.

Example 5: Find the derivative of. Now get a common denominator and simplify.

Ex 6: Compute the derivative of. Note: Therefore, we cannot use the properties of logs to bring the exponent down as a coefficient! Instead, we must use the Chain Rule.

YOU TRY… Find the derivative of each of the following: 1. 2.f(x) = ln (x 2 – 3) 5 3. f(x) = [ln (x 2 – 3)]

ln (x 2 – 3) 5 = 5 ln (x 2 – 3) = ln (3x +1) – ln (5x-2) Solution 1

Solution 4 Way 1

If we first simplify the given function using the laws of logarithms, then the differentiation becomes easier: Solution 4 Way 2

Solution 5

Closure Explain how to differentiate the following function both the hard way and the easy way.

Taking this derivative would involve the Power, Product, Quotient, and Chain (twice) Rules. Instead we can use a technique called logarithmic differentiation to simplify the process. LOGARITHMIC DIFFERENTIATION

Take the “ln” of both sides and use properties of logarithms to expand. Differentiate implicitly with respect to x.

Solve for dy / dx. Since we have an explicit expression for y, we can substitute and write:

1. Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify. 2. Differentiate implicitly with respect to x. 3. Solve the resulting equation for y’. 4. Replace y with f(x). STEPS IN LOGARITHMIC DIFFERENTIATION

Example 1: Differentiate. Solution: This problem could be easily done by multiplying the expression out.

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.

Chain Rule and Product Rule Ex 2.

Chain Rule and Quotient Rule Ex 3.

You Try… 3. Find dy/dx if.

Logarithmic differentiation is also used when the variable is in the base and the exponent.

Example 5: Differentiate.

Ex 6: Find an equation of the tangent line to the graph of at.

1. y = x x You Try… Differentiate each of the following functions. 2. y = (sinx) lnx

ln y = ln x x ln y = x ln x y = x x

Closure Explain the steps for the different methods used to differentiate the following functions. 1. y = x 4 2. y = x x