Partial Derivatives

(requires JavaScript)

1. Find the first partial derivatives of the function

   ${\displaystyle f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\frac{x-y}{x+y}}$

   $f_x<!--FUNCTION\; APPLICATION-->\left(x,y\right)=2y/{\left(x+y\right)}^2$, $f_y<!--FUNCTION\; APPLICATION-->\left(x,y\right)=-2x/{\left(x+y\right)}^2$

2. Find the first partial derivatives of the function

   ${\displaystyle f<!--FUNCTION\; APPLICATION-->\left(r,s\right)=r\ln <!--FUNCTION\; APPLICATION-->\left(r^2+s^2\right)}$

   ${\displaystyle f_r<!--FUNCTION\; APPLICATION-->\left(r,s\right)=\frac{2r^2}{r^2+s^2}+\ln <!--FUNCTION\; APPLICATION-->\left(r^2+s^2\right)}$, ${\displaystyle f_s<!--FUNCTION\; APPLICATION-->\left(r,s\right)=\frac{2rs}{r^2+s^2}}$

3. Find the first partial derivatives of the function

   ${\displaystyle f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=x^2{e}^{yz}}$

4. Find $f_y<!--FUNCTION\; APPLICATION-->\left(-6,4\right)$ if $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\sin <!--FUNCTION\; APPLICATION-->\left(2x+3y\right)$.
5. Verify that the conclusion of Clairaut's Theorem holds for the function $u<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x^4{y}^2-2x{y}^5$.

6. Find all second order partial derivatives of the function

   $z=f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x{e}^{xy}$

   $f_{xx}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=2y{e}^{xy}+x{y}^2{e}^{xy}$,
   $f_{yy}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x^3{e}^{xy}$,
   $f_{xy}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=f_{yx}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=2x{e}^{xy}+x^{2}y{e}^{xy}$.