Consider the function ƒ (x, y) = y \exp (a x)+x y +bx\ln y where a and b are constants.
a) Calculate \frac {\partial ƒ(x,y)}{\partial x}, \frac {\partial ƒ(x,y)}{\partial y} and dƒ(x, y).
b) Calculate \frac {\partial^2 ƒ(x,y)}{\partial x \partial y}.
The partial derivatives and differential of the function ƒ(x, y) = y \exp (a x) + x y + b x \ln y. are given by,
a)\frac {\partial ƒ(x,y)}{\partial x} = a y \exp (a x) + y + b \ln y .
\frac {\partial ƒ(x,y)}{\partial y} = \exp(a x) + x + \frac {b x}{ y} .
dƒ (x, y) = (ay \exp (ax) + y + b \ln y) d x +\biggl(\exp (a x)+x+\frac{b x}{y} \biggr)d y .
b) \frac {\partial^2 ƒ(x,y)}{\partial x \partial y} =a \exp (a x) + 1 +\frac {b }{ y} .