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When tasked with finding the derivative of a function that is a product of two separate functions, such as g(x) = e^x \cdot \arcsin x, we must make use of the product rule. This rule is a crucial part of calculus and it tells us that the derivative of the product f(x) \cdot g(x) is not simply the product of the derivatives of f(x) and g(x). Instead, the product rule states:

\[\begin{equation}(f \cdot g)' = f' \cdot g + f \cdot g'\end{equation}\]
This means we need to first take the derivative of the first function (f(x)) and multiply it by the second function (g(x)) as it is, then we add the product of the first function as it is with the derivative of the second function (g'(x)). Applying the product rule correctly is essential for accurately calculating the derivative of a product of functions.

The chain rule is another fundamental concept in calculus, particularly when dealing with composite functions. A composite function is made up when one function is applied within another, and in the case of differentiation, we cannot simply differentiate the outer function without considering the inner function. The chain rule tells us how to take the derivative of a composite function. It states that if you have a composite function h(x) = f(g(x)), then the derivative of h(x) with respect to x is:

\[\begin{equation} h'(x) = f'(g(x)) \cdot g'(x)\end{equation}\]
In other words, it's the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. When we differentiate the arcsine function or other trigonometric functions, the chain rule often comes into play when these functions are expressed in terms of another function.

Exponential functions are common in mathematics, particularly those with base e, the mathematical constant approximately equal to 2.71828. The beauty of the natural exponential function f(x)=e^x lies in its simplicity when differentiating. The derivative of e^x with respect to x is:

\[\begin{equation}f'(x) = e^x\end{equation}\]
This characteristic makes it an ideal and straightforward case when applying the product or chain rule, as there's no additional manipulation required unlike other functions. The exponential function does not change when differentiated, which is unique compared to most other types of functions.

The arcsine function represents the inverse of the sine function and is often represented as \arcsin x or \sin^{-1} x. It's not as straightforward to differentiate as the exponential function. Whenever we need to find the derivative of the arcsin function, we use a specific formula:

\[\begin{equation}(\arcsin u)' = \frac{u'}{\sqrt{1-u^2}}\end{equation}\]
In the case where u = x and u' = 1, this simplifies to:

\[\begin{equation}(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}\end{equation}\]
Understanding this derivative is important especially in cases where arcsin is part of a product of functions or a chain of functions. It's also essential to note that the domain of x for the arcsine function is limited to [-1, 1], since it represents angles whose sine gives x.