Calculus for Economics, September 25, 2017

I Find $f_x$ and $f_y$ for the following functions.

(1) $f(x,y)=(2x+3y)(3x+5y)$

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

(3) $f(x,y)=(2x+5y)^3$

(4) $f(x,y)=\left(\frac {2x+3y}{x+2y}\right)^2$

(5) $f(x,y)=ye^{x+y}$

II Find the stationary points of the following functions.

(1) $z=x^2+xy+y^2-4x-8y$

(2) $z=x^3+y^3-9xy+27$

(3) $z=x^2+xy-y^2-4x-2y$

(4) $z=x^2+4xy+2y^2-6x-8y$

(5) $z=x^3-xy-y^2$

(6) $z=e^{-x^2-y^2}(2x^2+y^2)$

(7) $z=(x^2+y^2)^2-2(x^2-y^2)$

(8) $z=x^3+y^3+6xy$