Are the following two derivatives same?

[sgsawant]

1. f_xy
2. f_yx


Are the above 2 derivatives equal, in general. Please explain if you know the answer.

Regards,

-sgsawant

 

[phyzguy]

Yes, in general if these two derivatives exist and are continuous, then they are equal. This is called "equality of mixed partial derivatives". I have also seen it called Clairaut's theorem, although supposedly it was first proved by Euler (like so much of the rest of mathematics).

 

[HallsofIvy]

As long as $f_{xy}$ and $f_{yx}$ are continuous in some neighborhood of a point, then, at that point, they are equal.

 

[Landau]

Even better:

Let U in R^2 be open, $f:U\to\matbb{R}$ partial differentiable w.r.t. both variables, and $D_1f$ partial differentiable w.r.t the second variable. Suppose further that (x,y) is in V, and $D_2D_1f$ is continuous at (x,y). Then $D_2f$ is partial differentiable w.r.t the first variable at (x,y), and

$$D_1D_2f(x,y)=D_2D_1f(x,y)$$.

i.e. we only need $D_2D_1f$ to exist and be continuous at some point in the interioir, this already implies that $D_1D_2f$ exists at that point and the two are equal.

 

[CellCoree]

I can say that the two derivatives, fxy and fyx, are not necessarily the same. In general, the order in which derivatives are taken can affect the result. This is known as the "order of differentiation" and it is an important concept in calculus.

To understand this concept, let's consider an example where f(x) = x^2. The first derivative of f(x) is 2x, and the second derivative is 2. Now, if we take the derivative of f(x) twice in a different order, we get:

1. fxy = (2x)' = 2
2. fyx = (x^2)' = 2x

As you can see, the two derivatives are not the same. In the first case, we took the derivative with respect to y first, and then with respect to x. In the second case, we took the derivative with respect to x first, and then with respect to y.

This example shows that the order of differentiation matters. In general, if a function has continuous second derivatives, then the mixed partial derivatives (like fxy and fyx) are equal. However, if the second derivatives are not continuous, then the mixed partial derivatives may not be equal.

In conclusion, the two derivatives, fxy and fyx, are not necessarily the same. The order in which derivatives are taken can affect the result, and this is an important concept to keep in mind when working with derivatives.