I worked on one my problem and I want you to check my work or give my some good idea for solution.
Let $f$ be function of two variables such that $f(x,y)=|xy|$.
a) Find all the point were partial derivates exist and compute partial derivates in it.
My work:
Our function f is symetirical so it doesn't matter which variables x or y sholud I look.
Derivation of $|x|$ is $\frac{x}{|x|}$ so our only problematic point wil be $(0,0)$ (in other point partial derivatives is easy to compute).
$$\frac{\partial f}{\partial x}(0,0)=lim_{h\rightarrow0}(\frac{f(h,0)}{h}=\frac{0}{h})=0$$. b) Find all the points were function f is differentiable.
My work: It is easy to see that only problematical point will be $(0,0)$. But we know all partial derivates (form part a) so we know Jacobian matrix and we can use definition to check differentiable.)
$$lim_(x,y)\rightarrow(0,0) \frac{|xy|}{\sqrt{x^2+y^2}}=0$$ (This is easy to check by switcing to polar coordinates.)
c) Find $Df(1,3)(5,7)$
My work: $$\frac{\partial f}{\partial x}(x,y)=\frac{x}{|x|}*|y|$$ $$\frac{\partial f}{\partial y}(x,y)=\frac{y}{|y|}*|x|$$
So by that we know how our Jacobian matrix looks like and we evalute her in point $(1,3)$ and after that we multiply our matrix at point $(1,3)$ by vector $(5,7)$.