Logarithmic Differentiation Lesson 3-8 Part 2 Logarithmic Differentiation

Objectives Know derivatives of regular and natural logarithmic functions Take derivatives using logarithmic differentiation

Logarithmic Differentiation Steps in Logarithmic Differentiation: Take natural log of both sides of an equation y = f(x) Use to laws of logs to simplify Differentiate implicitly with respect to x Solve the resulting equation for y’ (dy/dx) Substitute back in what y was in step 1.

Logarithmic Differentiation Example x² (5x² + 6x)³ y = ------------------- find y’ (7x³+ 3x²)³ ln y = 2 ln x + 3 ln (5x² + 6x) – 3 ln (7x³+ 3x²) 1 dy 1 10x + 6 21x² + 6x -- ----- = 2 • --- + 3 • ------------- – 3 • -------------- y dx x 5x² + 6x 7x³+ 3x² dy 1 10x + 6 21x² + 6x ----- = y(2 • --- + 3 • ------------- – 3 • -------------- ) dx x 5x² + 6x 7x³+ 3x² dy x² (5x² + 6x)³ 1 10x + 6 21x² + 6x ----- = ----------------- (2 • --- + 3 • ------------ – 3 • -------------- ) dx (7x³+ 3x²)³ x 5x² + 6x 7x³+ 3x²

Ugly chain & quotient rules Logarithmic Differentiation Example 1 Find derivative of the following: 1. y = (10x³ /  x + 1)4   Ugly chain & quotient rules or Logarithmic Differentiation ln y = ln ((10x³ /  x + 1)4 ) = 4[ln(10x³) – ln(x+1)½] = 4[ ln(10x³) – ½ ln(x+1)] y’ /y = 4[(30x² / 10x³) – ½ (1/(x+1)) ] y’ / y = (12/x) – (2/(x+1)) dy/dx = y (12/x) – (2/(x+1)) = ((10x³ /  x + 1)4 ) (12/x) – (2/(x+1))

Example 2 2. y = 6x Proving one of our rules Ln y = ln (6x) = x ln 6 y’ / y = ln 6 dy/dx = y (ln 6) = 6x (ln 6)

Logarithmic Differentiation Example 3 Find the derivatives of the following: Quotient Rule! or Logarithmic Differentiation 3. y= 1 - x² / (x + 1)⅔   ln y = ln (1 - x² / (x + 1)⅔) = ½ ln(1-x²) – ⅔ ln (x + 1) y’ / y = ½ (-2x/(1-x²)) - ⅔ (1 / (x + 1)) y’ = y [(-x/(1-x²)) - (2 / (3(x + 1))] y’ = 1 - x² / (x + 1)⅔ [(-x/(1-x²)) - (2 / (3(x + 1))]

Example 4 Find the derivatives of the following: 4. f(x) = xx   ln y = ln (xx) ln y = x ln x y’ / y = x(1/x) + (1)ln x product rule! y’ = y (1 + ln x) dy/dx = (xx) (1 + ln x)

Summary & Homework Summary: Homework: Logarithmic Differentiation can help solve complex derivatives involving products, quotients and exponents Homework: Pg 249: 7, 9, 11, 21, 24, 35, 40