函数$f(x,y) = x^2 + \tan\frac{xy}{x^2+y^2}$在$(1,0)$处需先考察偏导数定义或直接求导。注意到当$y=0$且$x\ne0$时，$\frac{xy}{x^2+y^2}=0$，故$f(x,0) = x^2 + \tan 0 = x^2$。因此$$\left.\frac{\partial f}{\partial x}\right|_{(1,0)} = \left.\frac{d}{dx}f(x,0)\right|_{x=1} = \left.\frac{d}{dx}(x^2)\right|_{x=1} = 2x\big|_{x=1} = 2.$$故答案为$2$。