1a.) $f_x = 2xy^2+3y$, $f_y=2x^2y+3y^2+3x$

1b.) $f_x = \cos(x)\sin(y)$, $f_y = \sin(x)\cos(y)$

1c.) $f_x = \frac{y}{xy} \sin(y^2)$, $f_y = \frac{2xy}{xy} \cos(y^2)$

1d.) $f_x = -\frac{1}{x^2}\sin(xy) + \frac{y}{x}\cos(xy)$, $f_y = \frac{x\cos(xy)}{x}$

1e.) $f_x = 0$, $f_y = 0$

1f.) $f_x = -y\sin(xy)\sqrt{\displaystyle\int_0^x \sin(t)dt}+\cos(xy) \frac{1}{2} \left( \displaystyle\int_0^x \sin(t) dt \right)^{-\frac{1}{2}} \sin(x)$, $f_y = -x\sin(xy) \sqrt{ \displaystyle\int_0^x \sin(t) dt}$

2a.) $f_{xx} = 0$, $f_{yy} = 0$, $f_{xy} = 3$

2b.) $f_{xx} = -\csc^2(x)$, $f_{yy} = -\frac{1}{y^2}$, $f_{xy} = 0$

2c.) $f_{xx} = -y^2 \csc^2(xy)$, $f_{yy} = -x^2 \csc^2(xy)$, $f_{xy} = \cot(xy) - xy \csc^2(xy)$

2d.) $f_{xx} = - \sin(x)$, $f_{yy}=0$, $f_{xy} = 0$

3a.) $3x^2yz^4$

3b.) $3x^2z$

3c.) $24xyz^3$

3d.) $0$

3e.) $6z^4$