Partial derivatives can exist while the function is not differentiable

The standard counterexample, worked — and why 'both partials exist' is the weakest hypothesis in multivariable calculus.

Step 1 / 4The candidate

Let

$f (x, y) = \frac{x y}{x ^{2} + y ^{2}} for (x, y) \neq = (0, 0), f (0, 0) = 0.$

Is $f$ differentiable at the origin?