The product rule and the power rule serve different purposes in differentiation. The power rule is specifically used for differentiating functions of the form x raised to a power. It provides a direct formula based on the exponent of the function. On the other hand, the product rule is employed when differentiating the product of two distinct functions. Instead of dealing with a single function raised to a power, the product rule considers two separate functions and their derivatives. In essence, while the power rule focuses on the exponent of a single function, the product rule addresses the derivatives of two functions being multiplied together.

The quotient rule is specifically designed for differentiating quotients, i.e., functions in fraction form where one function is divided by another. If a function is not in a fraction form, the quotient rule is not the most direct or efficient method for differentiation. Instead, other rules like the power rule, product rule, or chain rule might be more applicable. However, it's worth noting that any function can be expressed as a fraction by dividing it by 1, but applying the quotient rule in such cases would be redundant and unnecessary.

The chain rule is crucial because it allows for the differentiation of composite functions, where one function is embedded within another. In many real-world scenarios, functions are not always simple monomials or basic equations; they can be complex compositions of multiple functions. The chain rule provides a systematic approach to tackle such composite functions by differentiating the outer function while keeping the inner function unchanged, and then multiplying by the derivative of the inner function. This rule bridges the gap between simple differentiation rules and the complexities of composite functions, making it an indispensable tool in calculus.

The power rule is termed as such because it is used to differentiate functions that are raised to a "power" or exponent. The rule provides a straightforward method to find the derivative of monomials, which are single-term algebraic expressions with a coefficient and a variable raised to a non-negative integer exponent. The power rule simplifies the process by directly relating the exponent (or power) of the function to its derivative. Essentially, the "power" or exponent plays a central role in the differentiation process, hence the name "power rule".

Yes, while the basic differentiation rules like the power rule, product rule, quotient rule, and chain rule are powerful tools, they have their limitations. They are primarily designed for polynomial, rational, and certain algebraic functions. For functions that are transcendental (like exponential, logarithmic, or trigonometric functions), or for implicit functions, these basic rules might not be directly applicable. In such cases, additional rules and techniques, such as the rules for differentiating exponential or trigonometric functions, or methods like implicit differentiation, are required. It's essential to understand the nature of the function at hand and apply the most appropriate differentiation technique.