Question

Use implicit differentiation to find ∂z/∂x and ∂z/∂y. x2 + 4y2 + 7z2 = 1 ∂z/∂x = ________ ∂z/∂y = __________

Answer 1

Assuming $z=z(x,y)$, differentiating both sides wrt $x$ gives

$2x+14z\dfrac{\partial z}{\partial x}=0\implies\dfrac{\partial z}{\partial x}=-\dfrac x{17z}$

and wrt $y$ gives

$8y+14z\dfrac{\partial z}{\partial y}=0\implies\dfrac{\partial z}{\partial y}=-\dfrac4{7z}$