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Partial derivatives are the backbone of understanding how functions change with respect to their variables. In a multi-variable function like \(f(x, y)\), a partial derivative measures how the function changes in one variable while keeping the other constant.
For example, the partial derivative of \(f\) with respect to \(x\), denoted \(\frac{\partial f}{\partial x}\), identifies the rate at which \(f\) changes as \(x\) increases, assuming that \(y\) does not change. Similarly, \(\frac{\partial f}{\partial y}\) describes the rate of change of \(f\) with changes in \(y\), keeping \(x\) constant.
Calculating partial derivatives is essential when dealing with variable changes in multivariable calculus. They help us precisely estimate the change in function value due to small changes in the variables involved.

When we talk about the change in a function, we refer to how its value alters due to variations in its inputs. For a function \(f(x, y)\), if the inputs change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\), the change can be denoted as \(\Delta f\).
This change \(\Delta f\) can be viewed as the difference in function values: \(f(a+\Delta x, b+\Delta y) - f(a, b)\). However, calculating this directly might be complex for intricate functions.
Using methods like the total differential allows us to approximate \(\Delta f\) efficiently, especially when \(\Delta x\) and \(\Delta y\) are small. This approximation helps in simplifying the analysis and predictions of how a function behaves with changing inputs.

Approximating changes in a function is often necessary when dealing with small variations in the independent variables. This is where the concept of the total differential comes into play.
The total differential, \(df\), gives an approximate change of the function \(f\) based on its partial derivatives. The formula \(df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy\) provides a linear approximation of how the function changes when \(x\) changes by \(dx\) and \(y\) changes by \(dy\).
By substituting \(dx = \Delta x\) and \(dy = \Delta y\) into this formula, we can approximate \(\Delta f\), which represents the change in function value. This is notably useful in predicting small but significant adjustments in situations where exact computations might be too cumbersome. Thus, total differential allows for a streamlined means of estimating function changes effectively.