Logarithmic Differentiation Classroom

57.1 Introduction

Logarithmic differentiation is a powerful technique used to differentiate functions that are difficult to handle with standard differentiation rules. It is particularly useful for:

• Functions that are products or quotients of many terms.
• Functions raised to a power, where both the base and the exponent are variables (e.g., $f(x)^{g(x)}$).

The process involves taking the natural logarithm of both sides of the equation, using logarithm properties to simplify the expression, and then implicitly differentiating with respect to $x$.

57.2 Laws of Logarithms Review

Before applying logarithmic differentiation, it's crucial to be proficient with the fundamental laws of logarithms, as these simplify complex expressions into sums, differences, and simpler products.

• Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$
• Quotient Rule: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
• Power Rule: $\log_b(M^p) = p \log_b(M)$

These laws allow us to break down complicated products, quotients, and powers into simpler terms that are easier to differentiate.

57.3 Differentiating Log Functions

The basic derivatives of logarithmic functions are essential building blocks for logarithmic differentiation:

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$
• $\frac{d}{dx}(\ln u) = \frac{1}{u} \frac{du}{dx}$ (Chain Rule for natural log)
• $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$
• $\frac{d}{dx}(\log_a u) = \frac{1}{u \ln a} \frac{du}{dx}$ (Chain Rule for general log)

These derivatives, combined with the chain rule, form the basis for differentiating expressions after applying logarithmic properties.

57.4 Advanced Log Differentiation

This section applies logarithmic differentiation to more complex expressions, typically involving multiple factors in the numerator and denominator, often raised to various powers. The key steps are:

1. Take the natural logarithm of both sides.
2. Use log properties to expand the expression into sums and differences.
3. Differentiate both sides implicitly with respect to $x$.
4. Solve for $\frac{dy}{dx}$.
5. Substitute back the original expression for $y$.

57.5 Differentiating $[f(x)]^x$

One of the most important applications of logarithmic differentiation is for functions where both the base and the exponent are non-constant functions of $x$, i.e., $y = [f(x)]^{g(x)}$. Neither the power rule (which requires a constant exponent) nor the exponential rule (which requires a constant base) applies directly.

By taking the natural logarithm, the exponent can be brought down as a coefficient using the power rule of logarithms, transforming the problem into a product rule scenario.