# Partial derivatives (question I am grading).

• MathematicalPhysicist
In summary, the conversation discusses the function f:R^2->R and its partial derivative of 2nd order. It is stated that f_{xy}=0 for all (x,y)\in \mathbb{R}^2 if and only if f(x,y)=g(x)+h(y). The reasoning behind this is not clear, but it is suggested to use the mean value theorem to show that f_x=F(x) and f_y=G(y). The question then becomes how to show that F(x)=h'(x), which can possibly be done using the fundamental theorem of calculus.
MathematicalPhysicist
Gold Member
We have a function f:R^2->R and it has partial derivative of 2nd order.
Show that $f_{xy}=0 \forall (x,y)\in \mathbb{R}^2 \Leftrightarrow f(x,y)=g(x)+h(y)$

The <= is self explanatory, the => I am not sure I got the right reasoning.

I mean we know that from the above we have: $f_x=F(x)$ (it's a question before this one), but now besides taking an integral I don't see how to show the consequent.

Any thoughts how to show this without invoking integration?

My first thought is the mean value theorem which states that if $f'(x)=0$ then $f\equiv$constant., apply this to $x$ when $y$ yields $\partial_{y}f\equiv$constant, but this constant will be dependent on $y$ and therefor is a function of $y$.

Get the general idea now?

That's basically what I have written, so I know that f_x =F(x) and f_y=G(y). But this is where I am not sure how to procceed.

I mean I know that: if F(x)=h'(x), then G(x,y)= f(x,y)-h(x) then G_x=0 and then G=g(y).

The problem is how do I know that F(x)=h'(x) I am assuming that F is of this form, aren't I?

I think that as you effectively have the equation $f'(x)=g(x)$ then you want to show that there is a function $F(x)$ with the property $F'(x)=f(x)$, and I think that this follows from a version of the fundamental theorem of calculus. By extension $G(x)=F(x)+C$ will also satisfy this equation. I think you can say this because $f\in C^{1}$ as you have $f'(x)$.

## 1. What is a partial derivative?

A partial derivative is a mathematical concept used in calculus to determine the rate of change of a function with respect to one of its variables, while keeping all other variables constant.

## 2. What is the notation for a partial derivative?

The notation for a partial derivative is similar to that of a regular derivative, but with a subscript indicating which variable is being held constant. For example, the partial derivative of a function f(x,y) with respect to x is written as ∂f/∂x.

## 3. How do you calculate a partial derivative?

To calculate a partial derivative, you must first determine which variable is being held constant. Then, you can take the derivative of the function with respect to the variable that is changing, treating all other variables as constants.

## 4. What is the purpose of using partial derivatives?

Partial derivatives are used in various fields of science and mathematics to analyze how a multivariable function changes when only one variable is altered. This allows for a better understanding of complex systems and helps in making predictions and optimizations.

## 5. Can you give an example of a real-world application of partial derivatives?

One example of a real-world application of partial derivatives is in economics, where they are used to determine the marginal cost and marginal revenue of a company. This information can then be used to make decisions about production and pricing strategies.

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