How to examine differentiability of the function $$y+\cos{\sqrt[3]{(x^2+y^2)}}$$ at $$(0,0)$$?
Partial derivatives are not defined there. The differential of the function equals to $$1-\frac{2}{3}\frac{(x+y)\sin((x^2+y^2)^{(1/3)}))}{(x^2+y^2)^{(2/3)}}$$. What should I do to come to a conclusion on differentiability?
Your function is not differentiable at $(0,0)$. If it was, the function $x\mapsto0+\cos\left(\sqrt[3]{x^2}\right)$ would be differentiable at $0$. But it is not.