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Derive the total derivative of $f(x,y,z)=x\cdot y\cdot z$. I get how the total derivative works but I'm not so good at deriving things.

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The total derivative of $f$ is given by $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz,$

So you get: $df=yz\:dx+xz\:dy+xy\:dz.$

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In fact, a much more general statement can be made. Namely:

Theorem: Let $K:\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_p}\to\mathbb{R}^m$ be multilinear, then $K$ is differentiable everywhere on $\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_p}\approx \mathbb{R}^{n_1+\cdots+n_p}$ (where the identification is in the obvious way) and moreover

$D_K(a_1,\cdots,a_p)(x_1,\cdots,x_p)=\sum_{j=1}^{p}K(a_1,\cdots,x_j,\cdots,x_p)$

If you'd like to see a proof, just ask.