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?
1 Like
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.
2 Likes