1. Logarithmic Differentiation
Lesson 3-8 Part 2
Logarithmic Differentiation

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

3. 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.

4. 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 x x² + 6x
= 2 • • – 3 •
y dx x x² + 6x x³+ 3x²
dy x x² + 6x
----- = y(2 • • – 3 • )
dx x x² + 6x x³+ 3x²
dy x² (5x² + 6x)³ x x² + 6x
----- = (2 • • – 3 • )
dx (7x³+ 3x²)³ x x² + 6x x³+ 3x²

5. 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))

6. 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)

7. 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))]

8. 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)

9. 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