Consider the partial derivatives

Question

fx(x,y)=4x3y7−20x3y,
fy(x,y)=7x4y6−5x4.
Is there a function f
which has these partial derivatives?

If so, what is it?
f=


Are there any others?

Answer 1

Yes, there is a function f that has these partial derivatives. To find this function, we integrate each partial derivative with respect to its corresponding variable.

First, integrate fx(x,y) with respect to x:
f(x,y) = ∫(4x^3y^7 - 20x^3y) dx
= x^4y^7 - 5x^4y + g(y),

where g(y) is a function of y that arises from the integration constant.

Next, differentiate f(x,y) with respect to y to find fy(x,y):
fy(x,y) = ∂/∂y (x^4y^7 - 5x^4y + g(y))
= 7x^4y^6 - 5x^4 + g'(y).

To have fy(x,y) = 7x^4y^6 - 5x^4, we need g'(y) to be equal to zero. Therefore, we can choose g(y) = C, where C is a constant.

Thus, the function f(x,y) = x^4y^7 - 5x^4y + C satisfies both partial derivatives given.

There may be other functions that have the same partial derivatives, but they would differ by a constant term C.