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To understand a function's behavior in a multivariable context, we begin with first-order partial derivatives. These derivatives help us examine how the function changes as each variable varies, while keeping the other variables constant. In our exercise, we have a function \(f(x, y) = (x+y)^3\). Here, we need to find how \(f(x, y)\) changes with respect to \(x\) and \(y\) separately.

- **With respect to \(x\):** When calculating the partial derivative with respect to \(x\), we differentiate \(f\) while treating \(y\) as a constant. Thus, the derivative \(f_x = \frac{\partial f}{\partial x} = 3(x+y)^2\).
- **With respect to \(y\):** Similarly, to find the derivative with respect to \(y\), treat \(x\) as a constant. Then, the derivative \(f_y = \frac{\partial f}{\partial y} = 3(x+y)^2\).

These computations tell us that the rate of change of \(f\) with respect to \(x\) and \(y\) is the same in form, reflecting the symmetry in \((x+y)^3\).

Mixed partial derivatives involve taking derivatives with respect to different variables. In simpler terms, to find a mixed derivative, we first differentiate with respect to one variable, then take a derivative of that result with respect to the other variable. This process helps us examine how the variables jointly affect the function's rate of change.

In the given exercise, the function is \(f(x, y) = (x+y)^3\). Let's see how mixed partial derivatives are calculated:

• For \(f_{xy}\), we differentiate \(f_x = 3(x+y)^2\) with respect to \(y\), resulting in \(f_{xy} = 6(x+y)\).
  
• For \(f_{yx}\), we differentiate \(f_y = 3(x+y)^2\) with respect to \(x\), resulting in \(f_{yx} = 6(x+y)\).
  

We notice that \(f_{xy}\) and \(f_{yx}\) are equal, and this equality is often expected for "nice" functions—those that are continuous and have continuous second partial derivatives—which fits our scenario.

Partial differentiation is a powerful method used to study functions of several variables. Unlike ordinary differentiation, partial differentiation focuses on changing one variable at a time, holding others constant. This technique is essential in multivariable calculus, especially for functions where variables interact in complex ways.

Let's break it down using our function \(f(x, y) = (x+y)^3\):

• By partially differentiating with respect to \(x\) and \(y\), we gain insight into how each individual variable influences the function, as seen with \(f_x = 3(x+y)^2\) and \(f_y = 3(x+y)^2\).
  
• Second-order partial derivatives like \(f_{xx}\), \(f_{yy}\), \(f_{xy}\), and \(f_{yx}\) provide more detailed information on the curvature and direction of the function's change.
  

Partial differentiation is critical not only in mathematics but also in real-world applications such as physics and engineering, where it helps describe systems with multiple interacting factors. Understanding how to compute and interpret these derivatives allows for deeper insight into the behavior of functions in multidimensional spaces.