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Differentiation is a core concept in calculus that involves finding the rate at which a function changes at any given point. When dealing with a function of a single variable, the derivative is obtained by calculating the limit of the function's average rate of change over an infinitely small interval. This provides the instantaneous rate of change or the slope of the tangent line at any point on the function's graph.
In the context of the given exercise, differentiation is applied to find how the function \( f(x, y) = 3x^2y + 2 \) changes with respect to the variables \( x \) and \( y \). By treating each variable independently as a constant when finding the derivative with respect to the other, we use a specialized form of differentiation called partial derivatives. Thus, differentiation is combined with considerations of multivariable functions to understand changes in one direction while keeping the other constant.

Multivariable calculus extends the principles of single-variable calculus to functions of several variables. This is crucial for analyzing physical and geometric systems where multiple dimensions or variables are involved. Concepts like partial derivatives, gradients, and divergence are introduced to capture the behavior of multivariable functions.

For the function \( f(x, y) = 3x^2y + 2 \), multivariable calculus allows us to explore how changes in \( x \) or \( y \) affect the overall function. Partial derivatives provide tangible measurements of these rates of change across different axes or dimensions. Here, knowing the first partial derivatives \( \frac{\partial f}{\partial x} = 6xy \) and \( \frac{\partial f}{\partial y} = 3x^2 \) enables understanding how the function responds to changes in \( x \) independently from \( y \) and vice versa.

Derivative rules are essential guidelines that help simplify and standardize the differentiation process. In calculus, these include rules such as the power rule, product rule, quotient rule, and chain rule, among others. Each of these rules provides a method for handling different types of algebraic expressions when finding derivatives.

In our exercise, the differentiation of \( f(x, y) = 3x^2y + 2 \) employs the product rule indirectly because \( 3x^2y \) is a product of \( 3x^2 \) and \( y \). When differentiating with respect to \( x \), \( y \) is seen as a constant, simplifying the process to the power rule: the derivative of \( x^n \) is \( nx^{n-1} \). The differentiation with respect to \( y \), where \( x \) is constant, similarly makes use of the basic linearity and power rule. These rules guide mathematicians in accurately and efficiently calculating derivatives of more complex functions.