Question

Calcule pela definição as derivadas parcias de primeira ordem:
a) f(x,y)=x²-xy+2y²
b)f(x,y)= sqrt(3x-y)

Answer 1

Pela definição, temos que:

$\boxed{f(x,y) = \frac{\partial\ f}{\partial\ x}}$

$\boxed{f(x,y)=\frac{\partial\ f}{\partial\ y}}$

a)

 $f(x,y)=\frac{\partial\ f}{\partial\ x}\\\\ \frac{\partial\ f}{\partial\ x}=2x-y+0\\\\ \boxed{\frac{\partial\ f}{\partial\ x}=2x-y}$

$f(x,y)=\frac{\partial\ f}{\partial\ y}\\\\ \frac{\partial\ f}{\partial\ y}=0-x+4y\\\\ \boxed{\frac{\partial\ f}{\partial\ y}=4y-x}$

b)

$f(x,y)=\sqrt{3x-y}\\\\ f(x,y)=\frac{\partial\ f}{\partial\ x}\\\\ \boxed{\frac{\partial\ f}{\partial\ x}=\frac{3}{2\sqrt{3x-y}}}$

$f(x,y)=\frac{\partial\ f}{\partial\ y}\\\\\ \boxed{\frac{\partial\ f}{\partial\ y}=\frac{-1}{2\sqrt{3x-y}} }$