# A question about mixed partial derivative

• vacuum
In summary, there is a connection between differentiability and the equality of cross partials in a function f[x,y]. However, this is only true if the cross partials exist and the function is twice differentiable in both variables. Continuity of other partial derivatives may also be required. In general, the equality of cross partials is true for most functions unless they are extremely complex.

#### vacuum

Let there be a function f[x,y]: RxR->R

Is there any connection between the differentiability
(I am not sure that this is the right English term - I meant f[x,y]= a*dx+b*dy +something of smaller order)
and the equality fxy=fyx, where fxy means the derivative of f[x,y] first by y, and than by x ?

Originally posted by vacuum
Let there be a function f[x,y]: RxR->R

Is there any connection between the differentiability
(I am not sure that this is the right English term - I meant f[x,y]= a*dx+b*dy +something of smaller order)
and the equality fxy=fyx, where fxy means the derivative of f[x,y] first by y, and than by x ?

The idicated cross partials have to exist of course. Usually you wouls ensure that by requiring that f be twice differentiable in both variables.

Does that mean that double differentiability implies the fxy=fyx equality?(Or reverse plus continuity of other partial derivatives?)

Originally posted by vacuum

I think you have to have continuity of the second derivitives at any point where you want to show the cross partials are equal. Since this is just a feature of the cross partials, i.e. it's always true if you have the above conditions, you can't use it to prove the conditions exist.

Pretty generally, unless you have a fiendishly pathological function f, you can always take the equality of the cross partials for true.

Thanks again!
This really clarifies some things...

## 1. What is a mixed partial derivative?

A mixed partial derivative is a mathematical concept that measures the rate of change of a function with respect to two different independent variables at the same point. It is a combination of two partial derivatives, and it is used to calculate how much the function changes when both variables are changed simultaneously.

## 2. How is a mixed partial derivative calculated?

A mixed partial derivative is calculated by taking the derivative of a function with respect to one variable while treating the other variables as constants, and then taking the derivative of the resulting expression with respect to the other variable. This can be written mathematically as ∂²f/∂x∂y.

## 3. What is the difference between a mixed partial derivative and a regular partial derivative?

The main difference between a mixed partial derivative and a regular partial derivative is that in a mixed partial derivative, both variables are allowed to vary, while in a regular partial derivative, only one variable is allowed to vary at a time. This means that a mixed partial derivative takes into account the effects of changes in both variables on the function, while a regular partial derivative only considers the effect of one variable at a time.

## 4. When is it necessary to use mixed partial derivatives?

Mixed partial derivatives are commonly used in multivariable calculus, specifically in the study of functions of more than two variables. They are necessary when the rate of change of a function needs to be calculated with respect to multiple variables simultaneously, such as in optimization problems or when studying the behavior of a function in multiple dimensions.

## 5. What are some real-world applications of mixed partial derivatives?

Mixed partial derivatives have many applications in physics, engineering, and economics. For example, they are used in fluid mechanics to calculate the rate of change of velocity and temperature in a fluid at a specific point. They are also used in economic models to study the relationship between multiple variables, such as supply and demand, and how changes in one variable affect the other. Additionally, mixed partial derivatives are used in optimization problems to find the maximum or minimum values of a function in multiple dimensions.