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Understanding the concept of a partial derivative is fundamental in the study of multivariable calculus. In a function with several variables, a partial derivative represents the rate at which the function changes as one particular variable is varied, while keeping the other variables constant.

For instance, in a function like \( F(r, s, t) = r(9s^4 - t^5) \), the variable \( r \) can influence the function's value independent of \( s \) and \( t \). Calculating the partial derivative with respect to \( r \), denoted as \( F_r \), involves treating \( s \) and \( t \) as constants and differentiating only the \( r \) portion of the function. This concept is crucial for understanding how multivariable functions vary in different directions in their domain.

Multivariable calculus extends the principles and techniques of calculus to functions of several variables. Unlike single-variable calculus, which deals with functions of one variable, multivariable calculus involves working with space curves, surfaces, and functions defined in higher dimensions.

In the context of the exercise, the function \( F(r, s, t) \) depends on three variables, which means it lives in a three-dimensional space. Each partial derivative, such as \( F_{r} \), \( F_{rs} \), or \( F_{rst} \), tells us how the function behaves in relation to changes in one or more of these dimensions, respectively. Mixed partial derivatives, like \( F_{rst} \), involve taking derivatives with respect to different variables in sequence, showing us how mixed changes in variables affect the output. Multivariable calculus is fundamental to fields such as physics, engineering, and economics, where systems are influenced by several factors at once.

The power rule is a swift method to differentiate functions where the variable is raised to a power. In single-variable calculus, it states that if \( f(x) = x^n \), then the derivative, denoted as \( f'(x) \), is \( nx^{n-1} \). This rule can be directly applied to partial differentiation in functions of multiple variables.

When we partially differentiate a multivariable function, we apply the power rule with respect to the variable of interest while treating other variables as constants. For example, to find \( F_{rs} \) in the given function, the power rule is used to differentiate \( 9s^4 \) with respect to \( s \), yielding a result of \( 36s^3 \) as \( s \) is the variable of interest and the exponent \( 4 \) decrements by one. Understanding and applying the power rule correctly is essential in both single-variable and multivariable calculus for efficient computation of derivatives.