Partial Differentiation

Find the first order partial derivative with respect to y
 $f(x,y)=x^3y^2+3xe^y$  
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•  ^{$f_y(x,y)=3x^2y^2+3e^y$}  

•  $f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$  

•  $f_y(x,y)=2x^3y+3xe^y$  

•  $f_y(x,y)=6x^2+3e^y$  

Find the second order partial derivatives of   $f(x,y)=(3x+2y)^4$  

•  $f_{xx}(x,y)=12(3x+2y)^2\ ,\ \ f_{yy}(x,y)=24(3x+2y)^2$  

•  $f_{xx}(x,y)=36(3x+2y)^2\ \ ,\ \ f_{yy}(x,y)=8(3x+2y)^3$  

•  $f_{xx}(x,y)=24(3x+2y)^2\ \ ,\ \ f_{xy}(x,y)=32(3x+2y)$  

•  $f_{xx}(x,y)=108(3x+2y)^2\ \ ,\ \ f_{yy}(x,y)=48(3x+2y)^2$  

 What is the mixed, second order partial derivative of this function.  $f\left(x,y\right)=\sqrt{2x^2+y^2}$ 

•  $f_{xx}(x,y)=2(2x^2+y^2)^{-\frac{1}{2}}-4x^2(2x^2+y^2)^{-\frac{3}{2}}$  

•  $fxy(x,y)=-2xy\left(2x^2+y^2\right)^{-\frac{3}{2}}$  

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

•  $f_{xy}(x,y)=-4x^2(2x^2+y^2)^{-\frac{3}{2}}$  

if  $f(x,y)=x^3-2xy+xy^3+3y^2$  which of the following is true?

•  $f_x(x,y)=3x^2-2y+y^3$   $f_y(x,y)=-2x+6y+3xy^2$   $f_{xy}(x,y)=6x$  
  

•  $f_x(x,y)=-2x+6y+3xy^2$   $f_y(x,y)=3x^2-2y+y^3$   $f_{xy}(x,y)=-2+3y^2$  
  

•  $f_{xx}(x,y)=6x$   $f_{yy}(x,y)=6+6xy$  
   $f_{xy}(x,y)=-2+3y^2$  

•  $f_{xx}(x,y)=6+6xy$   $f_{yy}(x,y)=6x$  
   $f_{xy}(x,y)=-2+3y^2$  

Find  $f_x$  and  $f_y$  from the equation  $f\left(x,y\right)=\ \ln\left(\sqrt{x^2y^3}\right)$  

•  $f_x=\frac{1}{x}\ \ \ \ \ \ \ f_y=\frac{3}{2y}$  

•  $f_x=\frac{1}{xy}\ \ \ \ \ \ \ f_y=\frac{3x}{2y}$  

•  $f_x=\frac{1}{x}\ \ \ \ \ \ \ \ f_y=\frac{3x}{2}$  

•  $f_x=\frac{x^2}{y^3}\ \ \ \ \ f_y=\frac{3x^2}{2y^3}$  

Find the Partial derivative of f with respect to y for  $f\left(x,y\right)=\frac{y^2}{x+y}$  

•  $\frac{2x+y}{\left(x+y\right)^2}$  

•  $\frac{2xy+y^2}{\left(x+y\right)^2}$  

•  $\frac{2}{\left(x+y\right)^{ }}$  

•  $\frac{2x^2+y^2}{\left(x+y\right)^2}$  

Find  $\frac{\partial z}{\partial x}$  for  $z=x^2y^2e^{xy}$  

•  $xy^2e^{xy}\left(xy+2\right)$  

•  $xye^{xy}\left(xy+2\right)$  

•  $x^2y^2e^{xy}\left(xy+1\right)$  

•  $2xy^2e^{xy}\left(xy+1\right)$