편도함수

$f_x=2xe^{x^2+y^2}$, $f_y=2ye^{x^2+y^2}$

$f_{xx}=2e^{x^2+y^2}+4x^2e^{x^2+y^2}$, $f_{yy}=2e^{x^2+y^2}+4y^2e^{x^2+y^2}$

$f_{xy}=4xye^{x^2+y^2}$, $f_{yx}=4xye^{x^2+y^2}$

$f_x=e^{x+2y}$, $f_y=2e^{x+2y}$

$f_{xx}=e^{x+2y}$, $f_{yy}=4e^{x+2y}$

$f_{xy}=2e^{x+2y}$, $f_{yx}=2e^{x+2y}$

$f_x=\cos x\cos y$, $f_y=-\sin x\sin y$

$f_{xx}=-\sin x\cos y$, $f_{yy}=-\sin x\cos y$

$f_{xy}=-\sin y\cos x$, $f_{yx}=-\sin y\cos x$

$f_x=3(x+y)^2$, $f_y=3(x+y)^2$

$f_{xx}=6(x+y)$, $f_{yy}=6(x+y)$

$f_{xy}=6(x+y)$, $f_{yx}=6(x+y)$

$f_x=-5y^3\sin(5xy^3)$, $f_y=-15xy^2\sin(5xy^3)$

$f_{xx}=-25y^6\cos(5xy^3)$, $f_{yy}=-225x^2y^4\cos(5xy^3)-30xy\sin(5xy^3)$

$f_{xy}=-75xy^5\cos(5xy^3)-15y^2\sin(5xy^3)$, $f_{yx}=-75xy^5\cos(5xy^3)-15y^2\sin(5xy^3)$

$f_x=10x\cos(5x^2+2y^3)$, $f_y=6y^2\cos(5x^2+2y^3)$

$f_{xx}=10\cos(5x^2+2y^3)-100x^2\sin(5x^2+2y^3)$, $f_{yy}=12y\cos(5x^2+2y^3)-36y^4\sin(5x^2+2y^3)$

$f_{xy}=-60xy^2\sin(5x^2+2y^3)$, $f_{yx}=-60xy^2\sin(5x^2+2y^3)$

$f_x=\frac{2y^2}{\sqrt{4xy^2+1}}$, $f_y=\frac{4xy}{\sqrt{4xy^2+1}}$

$f_{xx}=-\frac{4y^4}{\sqrt{4xy^2+1}^3}$, $f_{yy}=-\frac{16x^2y^2}{\sqrt{4xy^2+1}^3}+\frac{4x}{\sqrt{4xy^2+1}}$

$f_{xy}=-\frac{8xy^3}{\sqrt{4xy^2+1}^3}+\frac{4y}{\sqrt{4xy^2+1}}$, $f_{yx}=-\frac{8xy^3}{\sqrt{4xy^2+1}^3}+\frac{4y}{\sqrt{4xy^2+1}}$

$f_x=2\sqrt{y}$, $f_y=5\sqrt{y}+\frac{2x+5y}{2\sqrt{y}}$

$f_{xx}=0$, $f_{yy}=\frac{5}{\sqrt{y}}-\frac{2x+5y}{4y^{3/2}}$

$f_{xy}=\frac{1}{\sqrt{y}}$, $f_{yx}=\frac{1}{\sqrt{y}}$

$f_x=-\frac{2x}{(x^2+y^2+1)^2}$, $f_y=-\frac{2y}{(x^2+y^2+1)^2}$

$f_{xx}=\frac{8x^2}{(x^2+y^2+1)^3}-\frac{2}{(x^2+y^2+1)^2}$, $f_{yy}=\frac{8y^2}{(x^2+y^2+1)^3}-\frac{2}{(x^2+y^2+1)^2}$

$f_{xy}=\frac{8xy}{(x^2+y^2+1)^3}$, $f_{yx}=\frac{8xy}{(x^2+y^2+1)^3}$

$f_x=5$, $f_y=-17$

$f_{xx}=0$, $f_{yy}=0$

$f_{xy}=0$, $f_{yx}=0$

$f_x=6x$, $f_y=0$

$f_{xx}=6$, $f_{yy}=0$

$f_{xy}=0$, $f_{yx}=0$

$f_x=\frac{2x}{(x^2+y)}$, $f_y=\frac{1}{(x^2+y)}$

$f_{xx}=-\frac{4x^2}{(x^2+y)^2}+\frac{2}{(x^2+y)}$, $f_{yy}=-\frac{1}{(x^2+y)^2}$

$f_{xy}=-\frac{2x}{(x^2+y)^2}$, $f_{yx}=-\frac{2x}{(x^2+y)^2}$

$f_x=\frac{1}{4xy}$, $f_y=-\frac{\ln x}{4y^2}$

$f_{xx}=-\frac{1}{4x^2y}$, $f_{yy}=\frac{\ln x}{2y^3}$

$f_{xy}=-\frac{1}{4xy^2}$, $f_{yx}=-\frac{1}{4xy^2}$

$f_x=5e^x\sin y$, $f_y=5e^x\cos y$

$f_{xx}=5e^x\sin y$, $f_{yy}=-5e^x\sin y$

$f_{xy}=5e^x\cos y$, $f_{yx}=5e^x\cos y$

$f(x,y) = x\sin y+ x+ C$, where $C$ is any constant.

$f(x,y) = \frac12x^2+xy+\frac12y^2+ C$, where $C$ is any constant.

$f(x,y) = 3x^2y-4xy^2+2y +C$, where $C$ is any constant.