Differentiability in multivariable calculus 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.

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