Can directional derivatives be written as linear combination of partial derivatives even if f is not differentiable?

There are many examples of functions — even discontinuous functions — that have all partial derivatives but for which that linearity formula fails. You can find many of them littered around in questions on this site. But here are a few. Set $f(0,0) = 0$ and for $(x,y)\ne (0,0)$ take \begin{align*} f(x,y) &= \frac{xy}{x^2+y^2} \\ f(x,y) &= \frac{xy^2}{x^2+y^4} \\ f(x,y) &= \frac{xy^2}{x^2+y^2} \end{align*} Find the gradient vector (hint: they'll all be $0$) at the origin, and compute the various directional derivatives.

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Calculus

Multivariable Calculus

Partial Derivative

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