Advanced Differentiation Techniques

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to find the derivative of functions that are products, quotients, or powers involving variable exponents. It is especially useful when the function is complicated or when both the base and the exponent are functions of x.

• Key Steps:

  1. Take the natural logarithm of both sides:

  2. Use logarithm properties to simplify the expression.

  3. Differentiate both sides with respect to x (implicit differentiation).

  4. Solve for and, if needed, substitute back for .

• Example:

  • Take of both sides:

  • Differentiating both sides:

    • Product rule:

  • So,

Derivatives of Logarithmic and Exponential Functions

When differentiating logarithmic and exponential functions, it is important to apply the chain rule and properties of logarithms and exponents.

• Derivative of :

• Derivative of :

• Example:

  • Let

  • So,

Derivatives of Functions Involving Natural Logarithms

When differentiating functions involving , use the chain rule and properties of logarithms.

• Example:

  • Use the quotient rule:

  • ,

  • ,

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

Combining Logarithmic Properties for Simplification

Complex logarithmic expressions can often be simplified using logarithmic identities before differentiating.

• Key Properties:

• Example:

  • Expand:

  • Differentiating:

Summary Table: Derivative Rules Used

Additional info: The above notes are based on the provided handwritten solutions, which cover problems involving logarithmic differentiation, the quotient rule, and the use of logarithmic and exponential derivative rules. The examples and explanations have been expanded for clarity and completeness.