Question

Let \(f\) be a function of two variables \(x\) and \(y .\) Describe the procedure for finding the first partial derivatives.

Short answer

The process to find the first partial derivatives of a function of two variables, \(x\) and \(y\), is to differentiate the function with respect to \(x\) and \(y\) separately, treating the other variable as a constant each time. These first partial derivatives represent the rates of change of the function in the directions of \(x\) and \(y\) respectively.

Step-by-step solution

Understand the Concept of Partial Derivative

A partial derivative of a function of multiple variables is the derivative of the function with respect to one of those variables, with the other variables held constant. In other words, it measures the rate at which the function changes with respect to that variable, ignoring all the others. For a function \(f(x, y)\), its first partial derivatives are usually denoted by \(\frac{\partial f}{\partial x}\) (with respect to x) and \(\frac{\partial f}{\partial y}\) (with respect to y).

Compute the Partial Derivative With Respect to \(x\)

To compute \(\frac{\partial f}{\partial x}\), consider all \(y'\)s as constants. Differentiate the function normally as done with a univariable function, treating the \(y'\)s as constants and applying the standard rules of differentiation. The result will be a function of both \(x\) and \(y\), because \(y\) was held constant and not eliminated during this process.

Compute the Partial Derivative With Respect to \(y\)

To compute \(\frac{\partial f}{\partial y}\), approach it in a similar way as in step 2 but this time, consider all \(x'\)s as constants. Differentiate the function normally as with a univariable function, treating the \(x'\)s as constants. Like before, the result will be a function of both \(x\) and \(y\), because \(x\) was held constant and not eliminated during this process.

Interpret the Results

The partial derivatives calculated represent the rates of change of the function \(f(x,y)\) in the directions of \(x\) and \(y\) respectively. If those rates of change were represented graphically, \(\frac{\partial f}{\partial x}\) would represent the slope of the function in the x direction at given points (x,y), and \(\frac{\partial f}{\partial y}\) would similarly represent the slope of the function in the y direction at those points.