I have the following function:
f[0, 0] = 0 f[x_, y_] := Exp[-(x^2 + y^2)^(-1)] How do I find its partial derivatives at any given point, including $(0,0)$? This doesn't work (obviously, I guess):
point={0,0} D[f[x, y], x] /. x -> point[[1]] /. y -> point[[2]] In case you have any problems, I recommend using the Derivative operator instead of D, since the latter works on expressions, while the former one can work on pure functions.
{Derivative[1, 0][f][x, y], Derivative[0, 1][f][x, y]} // TraditionalForm 
The subtlety here is that one cannot use it to find partial derivatives of f at {0,0}, e.g.
Derivative[0, 1][f][0, 0] Power::infy: Infinite expression 1/0 encountered. >> Infinity::indet: Indeterminate expression E^ComplexInfinity encountered. >> Power::infy: Infinite expression 1/0^2 encountered. >> Indeterminate
however, you can use Limit instead of Derivative:
Limit[ #, h -> 0] & /@ {( f[0 + h, 0] - f[0, 0] )/ h, ( f[0, 0 + h] - f[0, 0] )/ h} {0, 0}
In general, it works as we would like:
Limit[ #, h -> 0] & /@ { (f[x + h, y] - f[x, y])/ h, (f[x, y + h] - f[x, y])/ h} == { Derivative[1, 0][f][x, y], Derivative[0, 1][f][x, y] } True
One can also take advantage of FoldList to follow computing several limits, e.g. :
FoldList[ Limit, (f[x + h, y] - f[x, y])/h, {h -> 0, y -> 0, x -> 0}] // TraditionalForm 
We have shown that the partial derivative with respect to x is continuous at {0,0}. We can show much more; namely, that any derivative of any order vanishes at {0,0}. We find here the few first terms of a Taylor series expansion near {0,0} :
Limit[ Limit[ Normal @ Series[ f[x, y], {y, y0, 7}, {x, x0, 7}], x0 -> 0], y0 -> 0] 0
i.e. this function is not analytic at {0,0}, since the Taylor series expansion is identically 0, while the function f itself is not.
Limit and the definition of a derivative to do it in the general case? $\endgroup$ Limit of the difference ratio directly. $\endgroup$ f is a smooth function everywhere but it is not analytic in {0,0}. $\endgroup$ For evaluating derivatives for numerical arguments, one (somewhat neglected) function for the purpose is SeriesCoefficient[]. Of course, this produces derivatives scaled by factorials, so you have to multiply by the appropriate factorial factors to get the actual value of the derivative. For instance,
Derivative[3, 5][Function[{x, y}, Exp[-1/(x^2 + y^2)]]][1, 1] 117215/(256 Sqrt[E]) With[{p = 3, q = 5, x0 = 1, y0 = 1}, p! q! SeriesCoefficient[Exp[-1/(x^2 + y^2)], {x, x0, p}, {y, y0, q}]] 117215/(256 Sqrt[E]) 
f[x_, y_] := Exp[-(x^2 + y^2)^(-1)]point = {1, 1};D[f[x, y], x] /. {x -> point[[1]], y -> point[[2]]}This worked.. $\endgroup$