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**HallsofIvy** Since you have only two independent variables, x and y, you must have $\displaystyle \nabla f= \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$.

(I suspect that what you have written as "$\displaystyle \vec{V}f$" was intended to be $\displaystyle \nabla f$".)

So what are $\displaystyle \frac{\partial tan^{-1}\left(\frac{y}{x}\right)}{\partial x}$ and $\displaystyle \frac{\partial tan^{-1}\left(\frac{y}{x}\right)}{\partial y}$?

And what are $\displaystyle \frac{\partial cos\left(\frac{y}{x}\right)}{\partial x}$ and $\displaystyle \frac{\partial cos\left(\frac{y}{x}\right)}{\partial y}$?