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Understanding the natural logarithm (\text{ln}) is crucial for simplifying complex functions. The natural logarithm is the logarithm to the base of the mathematical constant \(e\) (approximately equal to 2.71828). In logarithmic differentiation, we take the natural logarithm of both sides of an equation to make differentiation easier. For example, when we have a function \(y = \frac{\big(x^2 + e^{2x}\big)^3 e^{-2x}}{\big(1 + x - x^2\big)^{2/3}}\), we apply \(\text{ln}\) to both sides: \(\text{ln}(y) = \text{ln}\big(\frac{\big(x^2 + e^{2x}\big)^3 e^{-2x}}{\big(1 + x - x^2\big)^{2/3}}\big)\).

Using properties of logarithms such as \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\) and \(\text{ln}(a^b) = b\text{ln}(a)\), we can break down and simplify complex expressions more easily. This aids in the next step, differentiation.

Differentiation is a fundamental concept in calculus where we find the derivative of a function, representing the rate of change. Logarithmic differentiation involves differentiating implicit functions and fractions more easily by first converting the problem using logarithms.

After applying the natural logarithm to our function and simplifying, we differentiate both sides with respect to \(x\). For example, if \(\text{ln}(y) = 3\text{ln}(x^2 + e^{2x}) - 2x - \frac{2}{3}\text{ln}(1 + x - x^2)\), we differentiate to get: \(\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}[3\text{ln}(x^2 + e^{2x})] - \frac{d}{dx}[2x] - \frac{d}{dx}[\frac{2}{3}\text{ln}(1 + x - x^2)]\).

This simplifies to \(\frac{dy}{dx} = y \big[ \frac{6x + 6e^{2x}}{x^2 + e^{2x}} - 2 - \frac{2 (1 - 2x)}{3 (1 + x - x^2)} \big]\).

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It has two main branches: differential calculus and integral calculus. Differential calculus is concerned with the concept of a derivative, while integral calculus deals with integration.

In this particular problem, we used differential calculus techniques, specifically logarithmic differentiation, to find the derivative of a complex function involving exponential, polynomial, and fractional terms. The steps were:

• Taking the natural logarithm of both sides of the equation to leverage logarithmic properties.
• Simplifying the logarithmic expression.
• Applying the rules of differentiation to each term.
• Substituting back the original function to express the final derivative.

This method is one of the many powerful tools in calculus used to simplify and solve problems that might otherwise be very difficult to handle.