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The limit definition of a derivative is fundamental to calculus and serves as the bedrock for understanding how derivatives are computed. It's written as:
\[\begin{equation}\begin{aligned}f'(x) = \lim_{{h\to0}} \frac{f(x+h) - f(x)}{h}\end{aligned}\end{equation}\]This expression tells us that the derivative of a function at a certain point is the slope of the tangent line to the function at that point. To find this slope, we consider the rate at which the function values change as the input values become infinitesimally close to each other.While the limit definition is precise, it can be quite complex and time-consuming to apply, especially for intricate functions. This complexity is a key reason alternative rules for differentiation were developed.

The power rule dramatically simplifies the process of differentiation when dealing with polynomial functions. It states that for a function \[\begin{equation}\begin{aligned}f(x) = x^n,\end{aligned}\end{equation}\]the derivative is\[\begin{equation}\begin{aligned}f'(x) = nx^{n-1}.\end{aligned}\end{equation}\]Not only is this method less computationally intensive than the limit definition, but it also allows for immediate computation of the derivative. This is incredibly useful when dealing with higher powers of x, as calculating limits can become unwieldy.

The chain rule is essential when tackling the derivatives of composite functions—functions that are composed of one function inside another. It is expressed as:
\[\begin{equation}\begin{aligned}\text{If } f(x) = g(h(x)), \text{ then } f'(x) = g'(h(x)) \cdot h'(x).\end{aligned}\end{equation}\]This rule enables us to differentiate the outer function (often referred to as the 'outer layer') first and then multiply it by the derivative of the inner function ('inner layer'). By breaking down complex functions into simpler components, we make the differentiation process more digestible and straightforward.

Whenever you encounter a function that is the product of two other differentiable functions, you'll need the product rule to find the derivative. The formula is given by:
\[\begin{equation}\begin{aligned}\text{If } f(x) = g(x)h(x), \text{ then } f'(x) = g'(x)h(x) + g(x)h'(x).\end{aligned}\end{equation}\]This allows you to differentiate each function separately and then combine them. It's crucial to respect the order since you first multiply the derivative of the first function by the second function, and then add the product of the first function and the derivative of the second function. The product rule helps in avoiding the lengthy process of using the limit definition for each term.

Similar to the product rule, the quotient rule is designed for functions that are ratios of two other differentiable functions. The quotient rule is a bit more involved and is written as:\[\begin{equation}\begin{aligned}\text{If } f(x) = \frac{g(x)}{h(x)}, \text{ then } f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h^2(x)}.\end{aligned}\end{equation}\]Using this rule, one would take the derivative of the numerator and multiply it by the denominator, subtract the product of the numerator and the derivative of the denominator, and then divide the entire result by the square of the denominator. It avoids the convoluted limit calculations that would otherwise be necessary for such functions.