1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.5 The Derivative of ln x

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 55  Derivatives for Natural Logarithms  Differentiating Logarithmic Expressions Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 55 Derivative Rules for Natural Logarithms

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 55 Differentiating Logarithmic ExpressionsEXAMPLE SOLUTION Differentiate. This is the given expression. Differentiate. Use the power rule. Differentiate ln[g(x)]. Finish.

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 55 Differentiating Logarithmic ExpressionsEXAMPLE SOLUTION The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point? This is the given function. Use the quotient rule to differentiate. Simplify. Set the derivative equal to 0.

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 55 Differentiating Logarithmic Expressions Set the numerator equal to 0. CONTINUED The derivative will equal 0 when the numerator equals 0 and the denominator does not equal 0. Write in exponential form. To determine whether the function has a relative maximum at x = 1, let’s use the second derivative. This is the first derivative. Differentiate.

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 55 Differentiating Logarithmic ExpressionsCONTINUED Simplify. Factor and cancel. Evaluate the second derivative at x = 1. Since the value of the second derivative is negative at x = 1, the function is concave down at x = 1. Therefore, the function does indeed have a relative maximum at x = 1. To find the y-coordinate of this point So, the relative maximum occurs at (1, 1).