Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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

share|cite|improve this answer

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


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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.