Question

preciso saber quais são as derivadas parciais de 1º ordem da função f(x,y) = (x²+3y).e^xy

Answer 1

$\displaystyle \frac{\partial f}{\partial x}= \frac{\partial }{\partial x}(x^2+3y)\cdot e^{xy}+(x^2+3y)\cdot\frac{\partial }{\partial x}e^{xy} \\ \\ \frac{\partial f}{\partial x}= 2xe^{xy}+(x^2+3y)\cdot ye^{xy}\\ \\ \\ \boxed{\frac{\partial f}{\partial x}= (2x+x^2y+3y^2)\cdot e^{xy}}$

$\displaystyle \frac{\partial f}{\partial y}= \frac{\partial }{\partial y}(x^2+3y)\cdot e^{xy}+(x^2+3y)\cdot\frac{\partial }{\partial y}e^{xy} \\ \\ \frac{\partial f}{\partial y}=3e^{xy}+(x^2+3y)\cdot xe^{xy}\\ \\ \boxed{\frac{\partial f}{\partial y}=(3+x^3+3xy)\cdot e^{xy}}$