3.6

Study Guide: Logarithmic Differentiation (3.6)

A crucial technique in calculus for differentiating complex functions.

Key Concepts:

  1. Logarithmic Functions: Simplify differentiation using properties of logarithms.

  2. Natural Logarithms: Essential for many calculus problems due to their frequent application.

  3. Logarithmic Differentiation: Especially useful for products and functions with exponents.

Short Cut Formulas:

  • For base ( a ):

    • If ( f(x) = \log_a(x) ), then ( f'(x) = \frac{1}{x \ln(a)} )

    • If ( f(x) = \ln(x) ), then ( f'(x) = \frac{1}{x} )

Important Properties:

  • The logarithmic function is defined only for positive values of x.

  • The base a must be greater than 0 and not equal to 1.

  • The derivative of a logarithmic function can be generalized using the change of base formula.

  • Product Property: ( \log_a(M) + \log_a(N) = \log_a(MN) )

  • Quotient Property: ( \log_a(M) - \log_a(N) = \log_a\left(\frac{M}{N}\right) )

  • Power Property: ( r \log_a(M) = \log_a(M^r) )

  • No simplification for ( \log_a(M + N) ) or ( \log_a(M - N) ).

Examples:

  1. Differentiate ( f(x) = \ln(1 - e^{-x}) )

    • Result: ( f'(x) = \frac{e^{-x}}{1 - e^{-x}} )

  2. Differentiate using Quotient Rule:

    • Find ( g'(x) ): Use provided formula with ( T = \ln(2) ) and ( B = 1 + \log_{12}(x) ).

  3. Differentiate ( h(x) = \ln[xx(x + 1)^2] )

    • Result: ( h'(x) = \frac{2}{x} + 2 \cdot \frac{1}{x + 1} )

Logarithmic Differentiation Technique:

  • E.g., Differentiate ( f(x) = (x + 1)^2(x + 2)^2(x + 3)^2 ) by considering ( \ln(y) ).

    • Final derivative includes evaluating several terms and simplifying.

Application:

  • Use logarithmic differentiation for functions like ( f(x) = \ln(\ln(x)) ) to handle products effectively.

Final Tip:

Understand the characteristics and properties of logarithmic functions and practice various examples to gain confidence before tests.This technique is vital for dealing with transcendental functions in more advanced calculus topics.