Does a differentiable function with $\nabla f(x,y) = (e^{x^2+\sin x}+y^5, -5xy^4 + \cos y)$ exist?

Question

Does a differentiable function $f:\mathbb{R^2} \to \mathbb{R}$ with

$$\nabla f(x,y) = (e^{x^2+\sin x}+y^5, -5xy^4 + \cos y)$$ exist?

I tried finding one by thinking of the integral, but since it's partial integration it makes it difficult due to the gradient

Answer 1

There is no such function, because if there was, then we would have to have$$\frac\partial{\partial y}\overbrace{(e^{x^2+\sin x}+y^5)}^{\phantom{\frac{\partial f}{\partial x}}=\frac{\partial f}{\partial x}}=\frac\partial{\partial x}\overbrace{(-5xy^4+\cos y)}^{\phantom{\frac{\partial f}{\partial y}}=\frac{\partial f}{\partial y}}.$$