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The slope of a curve at a point is a measure of how steep the curve is at that point. It's a fundamental concept in calculus that describes the rate of change of the function at a given point. When we talk about slope, we're generally referring to the derivative of the function at that point. For a curve described by an equation in two variables, like our example equation, \(x^{2/3} + y^{2/3} = 2\), finding the slope at a point requires a method called implicit differentiation since the equation isn't solved for \(y\) explicitly.

In this context, the slope at a point (1,1) provides us with an understanding of how the curve behaves in the immediate vicinity of this point. In our exercise, by determining the slope, we ascertain that the curve is descending at a 45-degree angle at the point (1,1), since the slope found is -1. This piece of information is invaluable for various applications, such as physics (for determining velocity or acceleration) or economics (to find out how one variable affects another).

Rational exponents are a way of expressing powers and roots in a unified manner. For instance, \(x^{2/3}\) denotes the cube root of \(x^2\), which can also be written as \(\sqrt[3]{x^2}\). Differentiating terms with rational exponents involves applying the standard power rule of differentiation. However, we must be careful with the negative exponents that arise after applying the rule.

In our exercise, differentiating \(x^{2/3}\) generates \(x^{-1/3}\), a negative exponent indicating the reciprocal of a cube root. The differentiation of rational exponents requires attention to ensure accurate simplification. Particularly, when we find the derivative at a point like (1,1), we capitalize on the property that the cube root of 1 is 1, simplifying our calculations considerably. This concept is not only essential for finding slopes of curves but is pervasive in all fields that utilize calculus.

The chain rule is a cornerstone of calculus, enabling us to differentiate compositions of functions. In implicit differentiation, it allows us to differentiate expressions where one variable is a function of another, such as the \(y\) term in our equation, \(y^{2/3}\). The chain rule dictates that we take the derivative of the outer function, in this case \(y^{2/3}\), as if \(y\) were just \(x\), and then multiply by the derivative of the inner function, which is \(dy/dx\).

In the step-by-step solution provided, we applied the chain rule by multiplying \((2/3)y^{-1/3}\) by \(dy/dx\). The result ties the rate of change of \(y\) to that of \(x\), a relationship that is crucial when the variables are interdependent. Mastering the chain rule is indispensable, as it finds application in various complex differentiation problems, such as those involving trigonometric functions, exponentials, or logarithms, not to mention its pivotal role in all forms of implicit differentiation.