Question

Determine as derivadas parciais indicadas.

Answer 1

a)

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

$\dfrac{\partial f}{\partial x}=\dfrac{1}{2}\dfrac{1}{\sqrt{x^2+y^2}}(2x)\ \therefore\ \boxed{f_x(x,y)=\dfrac{x}{\sqrt{x^2+y^2}}}$

$f_x(3,4)=\dfrac{3}{\sqrt{3^2+4^2}}=\dfrac{3}{\sqrt{25}}\ \therefore\ \boxed{f_x(3,4)=\dfrac{3}{5}}$

b)

$f(x,y,z)=\dfrac{x}{y+z}$

$\dfrac{\partial f}{\partial z}=x(-1)(y+z)^{-2}(1)\ \therefore\ \boxed{f_z(x,y,z)=-\dfrac{x}{(y+z)^2}}$

$f_z(3,2,1)=-\dfrac{3}{(2+1)^2}=-\dfrac{3}{9}\ \therefore\ \boxed{f_z(3,2,1)=-\dfrac{1}{3}}$