I am having issues with this mathematical concept, but i couldn't point out where, I've tried rereading thomas and stewart for more than 3 times already but still had no clue. I'll try to explain my thought process and will anyone of you please point out my mistakes?
given a function $f(x,y)=|xy|^{0.5}$, how do we determine if the partial derivative is defined at a point, lets say (0,0)? I've figured out there are 3 methods.
method 1:
$\frac{d}{dx} |xy|^{0.5} =\frac{(x^2*y^2)^{1/4}}{2x}$
then at $(0,0)$, it is undefined due to division-by-zero, therefore the P.D is undefined.
method2:
$f(x,0)=0$
$\frac{df}{dx}(0,0) = \frac{d}{dx}f(x,0)\vert_{y=0} = 0$ then at $(0,0)$, $f_x(0,0) = 0$. (P.D is defined)
method 3:
$\lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h} = 0$
therefore, $f_x(0,0) = 0$ (P.D is defined)
So what the differences between these method?? Did i apply those methods correctly? If so, why do i have contradicting solutions?
Thanks a lot for the help.