Solve Partial Derivatives: Find f(x,y)

[hotcommodity]

[SOLVED] Partial Derivatives

I'm having a bit of trouble on an old test problem. It states:

Determine if there is a function f(x, y) such that f_x(x, y) = ye^x + 1 and f_y(x, y) = e^x + cos(y). If such a function exists, find it.

I know that such a function exists because f_xy(x, y) = e^x, and f_yx(x, y) = e^x, thus f_xy = f_yx.

I'm having trouble finding the original function. I know that I would probably have to integrate somehow, but I'm not sure how to go about that with partial derivatives.

Any help is appreciated.

 

[quasar987]

integrate fx(x, y) with respect to x. This will tell you that f(x,y) is of the form e^x+x+C(y), where C(y) is some function of y. Do the same thing, but with fy(x, y).. you'll get f(x,y)=...+D(x) where CDx) is some function of x. Compare the two expressions of f(x,y) thus obtained, and conclude as to the value of C(y) and D(x), and hence of f(x,y)

 

[hotcommodity]

Ok, I found that f(x, y) = ye^x + x + sin(y), and it checks with the first order partial derivatives. Thank you very much :)

 

[HallsofIvy]

Plus a constant, of course.

 

[hotcommodity]

Right, I forgot that part, haha. Thanks :)