Question

If \(f\) is a function of \(x\) and \(y\) such that \(f_{x y}\) and \(f_{y x}\) are continuous, what is the relationship between the mixed partial derivatives? Explain.

Short answer

If \(f\) is a function of \(x\) and \(y\) such that the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\) are continuous, then according to Clairaut’s theorem, \(f_{xy} = f_{yx}\). That means the mixed partial derivatives are equal. They do not change with the order of differentiation.

Step-by-step solution

Definition of Mixed Partial Derivatives

In Calculus, a partial derivative of a function of several variables is its derivative with respect to one of those variables, where all the other variables are held constant. A mixed partial derivative, however, is when the derivative is taken more than once with respect to different variables. In other words, it is the derivative of a derivative. For example, if \(f\) is a function of two variables \(x\) and \(y\), the second order mixed partial derivatives are \(f_{x y} = \frac {\partial^2 f}{\partial x \partial y}\) and \(f_{xy} = \frac {\partial^2 f}{\partial y \partial x}\). Here, \(f_{x y} \) means the partial derivative of \(f\) with respect to \(x\) is taken first, and then the derivative of the result is taken with respect to \(y\). Similarly, \(f_{yx}\) means the partial derivative with respect to \(y\) is taken first, and then the derivative of that result is taken with respect to \(x\).

Theorem About The Equality of Mixed Partial Derivatives

There is a theorem in differential calculus called Clairaut’s theorem that states: If \(f\) is a function of \(x\) and \(y\) such that the second partial derivatives \(f_{xy}\) and \(f_{yx}\) are continuous, then \(f_{xy} = f_{yx}\). This means that the order of differentiation does not matter; the mixed partial derivatives are equal.

Explanation of The Result

On the basis of Clairaut’s theorem, it can be stated the mixed partial derivatives, \(f_{xy}\) and \(f_{yx}\) of a function \(f(x, y)\) which are continuous, are indeed equal. In other words, it does not matter whether you take the derivative first with respect to \(x\) and then with respect to \(y\) or vice versa, the result will still be the same.