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The function $f(x,y,z)$ is a differentiable function at $(0,0,0)$ such that $f_y(0,0,0)=f_x(0,0,0) = 0$ and $f(t^2,2t^2,3t^2)=4t^2$ for every t>0.
Define $u = (6/11,2/11,9/11)$, with the given about.
Is it possible to calculate $f_u(1,2,3)$ or $f_u(0,0,0)$, or $f_z(0,0,0)$?
If so, how do I calculate it?

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First of all, you assumption can be written as $f(s,2s,3s)=4s$ for every $s>0$: no need for $t^2$. Therefore, $\partial_x f(s,2s,3s)+2 \partial_y f(s,2s,3s)+3 \partial_z f(s,2s,3s)=4$ for every $s>0$. Hence, letting $s \to 0^+$, you find the value of $\partial_z f(0,0,0)$. This gives you a tool to compute $\partial_u f(0,0,0)$. Then choose $s=1$ to compute $\partial_u f(1,2,3)$.

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    Tha$n$ks a lot. I appreciate it2012-04-28