【微分積分】7-2-1 全微分可能の定義｜要点まとめ

偏導関数を計算すると
\(f_x(x,y)=2\)
\(f_y(x,y)=3\)
すなわち
\(\varepsilon(h,k)\)
\(=f(x+h,y+k)\)
\(\ \ \ -\{f(x,y)+f_x(x,y)h+f_y(x,y)k\}\)
\(=2(x+h)+3(y+k)+4\)
\(\ \ \ -\{(2x+3y+4)+2h+3k\}\)
\(=0\)
\(\displaystyle \lim_{(h,k)\to(0,0)}\frac{\varepsilon(h,k)}{\sqrt{h^2+k^2}}\)
\(\displaystyle =\lim_{(h,k)\to(0,0)}\frac{0}{\sqrt{h^2+k^2}}\)
\(\displaystyle =0\)
よって、全微分可能なので、
\(df=f_x(x,y)dx+f_y(x,y)dy\)
\(=2dx+3dy\)

偏導関数を計算すると
\(f_x(x,y)=4x+3y^2\)
\(f_y(x,y)=6xy\)
すなわち
\(\varepsilon(h,k)\)
\(=f(x+h,y+k)\)
\(\ \ \ -\{f(x,y)+f_x(x,y)h+f_y(x,y)k\}\)
\(=2(x+h)^2+3(x+h)(y+k)^2\)
\(\ \ \ -\{(2x^2+3xy^2)+(4x+3y^2)h+6xyk\}\)
\(=2h^2+6yhk+3xk^2+3hk^2\)
\(\displaystyle \lim_{(h,k)\to(0,0)}\frac{\varepsilon(h,k)}{\sqrt{h^2+k^2}}\)
\(\displaystyle =\lim_{(h,k)\to(0,0)}\frac{2h^2+6yhk+3xk^2+3hk^2}{\sqrt{h^2+k^2}}\)
\(\displaystyle =0\)
よって、全微分可能なので、
\(df=f_x(x,y)dx+f_y(x,y)dy\)
\(=(4x+3y^2)dx+6xydy\)