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I have this function $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2} & \mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y) = (0,0) \end{array} \right. $$ And I want to find the directional derivative in the $(1,1)$ direction at $(0,0)$, so using the limit definition this gives me $$ \begin{align*} D_vf(0,0) = \lim_{t\to 0}\frac{f((0,0) + t(1,1)) - f(0,0)}{t} = \lim_{t\to 0}\frac{t^3}{2t^2 t} = \frac{1}{2} \end{align*} $$ But wolfram alpha is giving me $0$ as a result Wolfram alpha result
Is there something wrong with my procedure?
Any help appreciated :)

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up vote 9 down vote accepted

You are right and Wolfram is wrong. What happens is that the computer is using the gradient to calculate the directional derivative. But the formula to calculate the directional derivative using the gradient uses the chain rule, which assumes the function to be differentiable. Your function is not differentiable at $(0,0)$, so the gradient cannot be used to find the directional derivatives there.

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