題目 6. (10 points) Consider the function f(x,y)={2x3+xyx2+y2,(x,y)≠(0,0)0,(x,y)=(0,0).f(x,y)= \begin{cases} \dfrac{2x^3+xy}{x^2+y^2}, & (x,y)\ne(0,0) \\ 0, & (x,y)=(0,0). \end{cases}f(x,y)=⎩⎨⎧x2+y22x3+xy,0,(x,y)=(0,0)(x,y)=(0,0). Find ∂f∂x\dfrac{\partial f}{\partial x}∂x∂f at (0,0)(0,0)(0,0) if existed. If it does not exist, explain why. 解答 解法一 由偏導數定義，f(h,0)=2hf(h,0)=2hf(h,0)=2h，故 fx(0,0)=limh→02h−0h=2.f_x(0,0)=\lim_{h\to0}\frac{2h-0}{h}=\boxed2.fx(0,0)=h→0limh2h−0=2.
