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Define function $f: \mathbb{R} ^3 \to \mathbb{R}$ as \begin{equation} f(x,y,z) = x^4 + y^4 + z^4 - 4xyz \end{equation} Show that $f$ is differentiable at the point $(1,1,1)$.

Solution: I thought about using the good old \begin{equation} \lim _{\bf{h} \to \bf{0}} \frac{|f(\bf{a}+\bf{h}) - f(\bf{a}) - \nabla f(\bf{a}) \cdot \bf{h}|}{||\bf{h}||} = 0 \end{equation} But that proved to be difficult so now I'm back to square one. Are there any alternative ways to evaluate differentiability at a point?


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up vote 3 down vote accepted

You should check that each of the partial derivatives exist and are continuous. This will give you that the function itself is differentiable.

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In case it helps: the sums and products of differentiable functions are again differentiable. This gives differentiability of any polynomial! Other arguments help when there are some indeterminacies.

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