# Derive the total derivative for a function

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.

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