pochodna cząstkowa 2 rzędu
: 12 sie 2010, o 13:19
oblicz wszystkie pochodne cząstkowe 2 rzędu podanych funkcji:
do sprawdzenia
1.
\(\displaystyle{ f(x,y)=ln(x^{2}+4y)+x^{2}y}\)
\(\displaystyle{ f'x= \frac{1}{x^{2}+4y} \cdot 2x+2xy}\)
\(\displaystyle{ f'y= \frac{1}{x^{2}+4y} \cdot 4+x^{2}}\)
\(\displaystyle{ f'xx= ( -\frac{1}{(x^{2}+4y)^{2}} \cdot 2x) \cdot 2x+\frac{1}{(x^{2}+4y)^{2}} \cdot 2+2y}\)
\(\displaystyle{ f'yy=(-\frac{1}{(x^{2}+4y)^{2}} \cdot 4) \cdot 4}\)
\(\displaystyle{ f'xy=(-\frac{1}{(x^{2}+4y)^{2}} \cdot 4) \cdot 2x+2x}\)
\(\displaystyle{ f'yx=(-\frac{1}{(x^{2}+4y)^{2}} \cdot 2x) \cdot 4+2x}\)
2.
\(\displaystyle{ f(x,y)=sin(x^{2}+y^{2)})}\)
\(\displaystyle{ f'x=cos(x^{2}+y^{2)}) \cdot 2x}\)
\(\displaystyle{ f'y=cos(x^{2}+y^{2)}) \cdot 2y}\)
\(\displaystyle{ f'xx=[-sin(x^{2}+y^{2)}) \cdot 2x] \cdot 2x+cos(x^{2}+y^{2)}) \cdot 2}\)
\(\displaystyle{ f'yy=[-sin(x^{2}+y^{2)}) \cdot 2y] \cdot 2y+cos(x^{2}+y^{2)}) \cdot 2}\)
\(\displaystyle{ f'xy=[-sin(x^{2}+y^{2)}) \cdot 2y] \cdot 2x}\)
\(\displaystyle{ f'yx=[-sin(x^{2}+y^{2)}) \cdot 2x] \cdot 2y}\)
do sprawdzenia
1.
\(\displaystyle{ f(x,y)=ln(x^{2}+4y)+x^{2}y}\)
\(\displaystyle{ f'x= \frac{1}{x^{2}+4y} \cdot 2x+2xy}\)
\(\displaystyle{ f'y= \frac{1}{x^{2}+4y} \cdot 4+x^{2}}\)
\(\displaystyle{ f'xx= ( -\frac{1}{(x^{2}+4y)^{2}} \cdot 2x) \cdot 2x+\frac{1}{(x^{2}+4y)^{2}} \cdot 2+2y}\)
\(\displaystyle{ f'yy=(-\frac{1}{(x^{2}+4y)^{2}} \cdot 4) \cdot 4}\)
\(\displaystyle{ f'xy=(-\frac{1}{(x^{2}+4y)^{2}} \cdot 4) \cdot 2x+2x}\)
\(\displaystyle{ f'yx=(-\frac{1}{(x^{2}+4y)^{2}} \cdot 2x) \cdot 4+2x}\)
2.
\(\displaystyle{ f(x,y)=sin(x^{2}+y^{2)})}\)
\(\displaystyle{ f'x=cos(x^{2}+y^{2)}) \cdot 2x}\)
\(\displaystyle{ f'y=cos(x^{2}+y^{2)}) \cdot 2y}\)
\(\displaystyle{ f'xx=[-sin(x^{2}+y^{2)}) \cdot 2x] \cdot 2x+cos(x^{2}+y^{2)}) \cdot 2}\)
\(\displaystyle{ f'yy=[-sin(x^{2}+y^{2)}) \cdot 2y] \cdot 2y+cos(x^{2}+y^{2)}) \cdot 2}\)
\(\displaystyle{ f'xy=[-sin(x^{2}+y^{2)}) \cdot 2y] \cdot 2x}\)
\(\displaystyle{ f'yx=[-sin(x^{2}+y^{2)}) \cdot 2x] \cdot 2y}\)