Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = y \cos ( x y )$ and $\frac { \partial f } { \partial y } = x \cos ( x y )$
A) $f ( x , y ) = \cos ( x y )$
B) $f ( x , y ) = \sin ( x y )$
C) $f ( x , y ) = - \sin ( x y )$
D) $f ( x , y ) = - \cos ( x y )$

Question

Quizplus · Accepted Answer

To find a function $f(x, y)$ with the given partial derivatives, integrate $\frac{\partial f}{\partial x} = y\cos(xy)$ with respect to $x$ and $\frac{\partial f}{\partial y} = x\cos(xy)$ with respect to $y$. Both integrations lead to $f(x, y) = \sin(xy)$ plus a constant, but since we're looking for the general form, the constant is not shown in the options. Therefore, the correct answer is $f(x, y) = \sin(xy)$.

Find a Function with the Given Partial Derivatives ∂f∂x=ycos⁡(xy)\frac { \partial f } { \partial x } = y \cos ( x y )∂x∂f​=ycos(xy)

Find a Function with the Given Partial Derivatives $\frac { \partial f } { \partial x } = y \cos ( x y )$