《문제》편도함수

1. 다음 함수의 원점 \((0,\,0)\)에서의 편미분계수를 구하여라.

(1) \(f(x,\,y)=\sqrt{|xy|}\)          (2) \(f(x,\,y)=\sqrt{x^2+y^2}\)          (3) \(f(x,\,y)=\begin{cases}\frac{xy}{\sqrt{x^2+y^2}}&(x,\,y)\ne(0,\,0)\\0&(x,\,y)=(0,\,0)\end{cases}\)

<풀이>

(1) \(f_x(0,\,0)=\lim_{h\to0}\frac{f(h,\,0)-f(0,\,0)}{h}=0,\,f_y(0,\,0)=\lim_{h\to0}\frac{f(0,\,h)-f(0,\,0)}{h}=0\)

(2) \(f_x(0,\,0)=\lim_{n\to\pm0}\frac{f(h,\,0)-f(0,\,0)}{h}=\pm1,\,f_y(0\,0)=\lim_{h\to\pm0}\frac{f(0,\,h)-f(0,\,0)}{h}=\pm1\)

     ∴ 원점에서는 편미분계수가 존재하지 않는다.

(3) \(f_x(0,\,0)=\lim_{h\to0}\frac{f(h,\,0)-f(0,\,0)}{h}=0,\,f_y(0,\,0)=\lim_{h\to0}\frac{f(0,\,h)=f(0,\,0)}{h}=0\)

2. 다음 함수를 편미분하여라.

(1) \(f(x,\,y)=x^2+3xy^2-2x+4y+3\)

(2) \(f(x,\,y)=x^3+3x^2y-y^3\)          (3) \(f(x,\,y)=2x+ye^{-x}\)

(4) \(f(x,\,y)=\sqrt{x^2+xy+y^2}\)         (5) \(f(x,\,y)={\rm Sin}^{-1}{x\over y}\)

(6) \(f(x,\,y)={\rm Tan}^{-1}\frac{x+y}{x-y}\)                (7) \(f(x,\,y)=e^{ax}\cos{by}\)

<풀이>

(1) \(f_x(x,\,y)=2x+3y^2-2,\,f_y(x,\,y)=6xy+4\)

(2) \(f_x(x,\,y)=3x^2+6xy,\,f_y(x,\,y)=3x^2-3y^2\)

(3) \(f_x(x,\,y)=\)