Advanced Differentiation Techniques

24. Differentiation of Logarithmic Functions with Composite Arguments

This section explores the differentiation of logarithmic functions where the argument is a product or composition of exponential and trigonometric functions.

• Key Point 1: To differentiate , use the change of base formula and the chain rule.

• Key Point 2: The derivative of is .

• Example:

  • Given

  • Let

  • Then

  • Compute using the product rule:

  • So,

28. Differentiation of Logarithmic Quotients

This section covers the differentiation of functions involving quotients of logarithmic and polynomial expressions.

• Key Point 1: Use the quotient rule: .

• Key Point 2: The derivative of is .

• Example:

  • Given

  • Let ,

  • ,

  • Apply the quotient rule:

  • Simplify numerator:

  • For the second derivative, apply the quotient rule again to .

44. Logarithmic Differentiation of Complex Rational Functions

This section demonstrates the use of logarithmic properties and differentiation for functions involving quotients and products inside a logarithm.

• Key Point 1: Use logarithmic identities to simplify: and .

• Key Point 2: The derivative of is .

• Example:

  • Given

  • Expand using logarithm rules:

  • Differentiate:

  • Or,

52. Logarithmic Differentiation of Exponential Trigonometric Functions

This section covers the differentiation of functions where both the base and the exponent are functions of , using logarithmic differentiation.

• Key Point 1: For , take the natural logarithm of both sides: .

• Key Point 2: Differentiate both sides implicitly, then solve for .

• Example:

  • Given

  • Take of both sides:

  • Differentiating both sides:

  • So,

Additional info: The problems above demonstrate the use of the product rule, quotient rule, chain rule, and logarithmic differentiation, which are essential techniques for differentiating complex functions in calculus.