Find the First Partial Derivatives of the Function f(x, y)

To find the first partial derivatives of a function f(x, y), we need to differentiate the function with respect to each variable while treating the other variable as a constant.

Let’s denote the partial derivative with respect to x as f_x(x, y) and the partial derivative with respect to y as f_y(x, y).

For example, if we have a function like:

• f(x, y) = x^2y + 3xy^2

To find f_x(x, y), we treat y as a constant and differentiate with respect to x:

• f_x(x, y) = 2xy + 3y^2

Then, to find f_y(x, y), we treat x as a constant and differentiate with respect to y:

• f_y(x, y) = x^2 + 6xy

Thus, the first partial derivatives are:

• f_x(x, y) = 2xy + 3y^2
• f_y(x, y) = x^2 + 6xy

This method can be applied to any function of two variables to find the first partial derivatives.