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The chain rule is a fundamental tool in calculus that allows us to find the derivative of a composite function. In essence, if we have two functions, say f and g, and they are combined such that one function takes the output of the other as its input (forming f(g(x))), then the chain rule helps us to differentiate this composition with respect to x.

To apply the chain rule, we differentiate the outer function f with respect to its input, which is g(x), and multiply this by the derivative of the inner function g with respect to x. Mathematically, this is represented as: \
\( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

This procedure is crucial when the function we're dealing with is not just a simple polynomial or exponential function but a more complex blend of functions.

Exponential functions, represented as e^x, where e is the Euler's number (approximately 2.71828), are unique in calculus due to their special properties. Their rate of change, or derivative, is the same as the function value itself. This means that for any exponential function e^x, the derivative with respect to x is e^x itself.

In mathematical terms, if we represent our function as g(x) = e^x, then: \
\( g'(x) = \frac{d}{dx}e^{x} = e^{x} \)

This property makes the exponential function very important and widely applicable in various fields such as physics, economics, and biology. When it comes to differentiation, this simplicity provides a straightforward path when we're using it within the chain rule.

Inverse trigonometric functions, like arctan(x), pose a different sort of challenge in calculus. To find the derivative of such functions, specific rules apply. For arctan(x), which is the inverse of the tangent function, the rate of change is given by the formula: \
\( \frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^{2}} \)

This formula allows us to find the slope of the tangent line to the curve y = arctan(x) at any given point. It's particularly useful when dealing with inverse trigonometric functions as part of a composite function, where we apply this derivative as part of the chain rule. Inverse trigonometric functions are often encountered in scenarios involving angles or when a function's range is limited within certain bounds and have various applications in geometry, physics, and engineering.