Hi, I have a quick question…

If a function is totally differentiable, is that the same thing as saying that all its partial derivatives exist?

If not, can you give me an example of a function where all of its partial derivatives exist, but it’s not totally differentiable? Thanks. None of my Calculus textbooks have info about this.

Yes.

For example, if z = f\left(x,y\right) then its total differential is:

dz\ =\ \frac{∂f}{∂x}dx\ +\ \frac{∂f}{∂y}dy

What brings this question to mind?

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Thanks! I was reading a definition for the delta method for finding the MLE of a function of multivariate random variable, and in its definition, it requires that a function be totally differentiable. The definition is long and confusing, but the examples given are pretty straightforward.

If the function has a cusp (e.g. y=x^(2/3)), dy/dx does exist, but not for x=0.

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