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?
Thanks.