題目 2. (10 points) Evaluate ∂f∂x∣(0,0)\left.\dfrac{\partial f}{\partial x}\right|_{(0,0)}∂x∂f(0,0) and ∂f∂y∣(0,0)\left.\dfrac{\partial f}{\partial y}\right|_{(0,0)}∂y∂f(0,0) for f(x,y)={3x4+xy22x3+4xy+y,(x,y)≠(0,0)0,(x,y)=(0,0).f(x,y)= \begin{cases} \dfrac{3x^4+xy^2}{2x^3+4xy+y}, & (x,y)\ne(0,0) \\ 0, & (x,y)=(0,0). \end{cases}f(x,y)=⎩⎨⎧2x3+4xy+y3x4+xy2,0,(x,y)=(0,0)(x,y)=(0,0). 解答 解法一 依偏導定義， fx(0,0)=limh→0f(h,0)h=limh→032hh=32.f_x(0,0)=\lim_{h\to0}\frac{f(h,0)}h =\lim_{h\to0}\frac{\frac32h}{h} =\boxed{\frac32}.fx(0,0)=h→0limhf(h,0)=h→0limh23h=23. 又 f(0,k)=0f(0,k)=0f(0,k)=0，故 fy(0,0)=limk→0f(0,k)k=0.f_y(0,0)=\lim_{k\to0}\frac{f(0,k)}k=\boxed0.fy(0,0)=k→0limkf(0,k)=0.
