Advanced Differentiation Techniques

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to differentiate functions where both the base and the exponent are variable, or when the function is a product or quotient of several functions. This method often simplifies the differentiation process by taking the natural logarithm of both sides.

• Key Steps:

  1. Take the natural logarithm of both sides: if , then .

  2. Use logarithm properties to simplify the expression.

  3. Differentiating both sides with respect to .

  4. Solve for .

• Useful for: Functions of the form , products, and quotients.

Example 1: Differentiating a Logarithmic Function (Problem 24)

Given:

• Step 1: Use the change of base formula: .

• Step 2: Differentiate using the chain rule:

• Key Points:

  • Apply the product and chain rules carefully.

  • Remember to multiply by in the denominator due to the change of base.

Differentiation of Logarithmic and Exponential Functions

When differentiating functions involving logarithms and exponentials, use the following rules:

• Derivative of :

• Derivative of :

• Product Rule:

• Quotient Rule:

Example 2: Differentiating a Logarithmic Quotient (Problem 28)

Given:

• First Derivative:

• Second Derivative:

Additional info: The second derivative requires the product and chain rules, and further simplification is possible.

Logarithmic Properties in Differentiation

Logarithmic properties can simplify complex expressions before differentiation:

Example 3: Differentiating a Complex Logarithmic Function (Problem 44)

Given:

• Step 1: Expand using logarithm properties:

• Step 2: Differentiate both sides:

Differentiation of Functions with Variable Exponents

For functions of the form , logarithmic differentiation is especially useful.

Example 4: Differentiating (Problem 52)

• Step 1: Take the natural logarithm of both sides:

• Step 2: Differentiate both sides:

• Step 3: Solve for :

Summary Table: Differentiation Rules Used

Additional info: The table summarizes the main differentiation rules applied in the problems above.