Solution assignment 05 Partial differentiation

Calculate the first, second and mixed partial derivatives of the following function.

$f(x,y)=3e^{x^2}y-xe^y$

For this function we need to apply the chain rule and the product rule.

$f_x=3\cdot2x\cdot{e^{x^2}}y-e^y=6xye^{x^2}-e^y$

$f_y=3e^{x^2}-xe^y$

$f_{xx}=6ye^{x^2}+12x^2ye^{x^2}$ (Here we have used the product rule)

$f_{yy}=-xe^y$

$f_{xy}=f_{yx}=6xe^{x^2}-e^y$

Again we notice that the mixed derivatives are equal.