Question

15. The equation below defines z as a differentiable function of x and y. Find the value of dz/dy at the point (1, 1, 1).

Answer 1

Isolate the term with $z^2$.

$x^2 - 5y^2 + xyz^2 = y - 4 \implies xyz^2 = -x^2 + 5y^2 + y - 4$

Differentiate both sides with respect to $y$.

$\dfrac{\partial(xyz^2)}{\partial y} = \dfrac{\partial(-x^2 + 5y^2 + y - 4)}{\partial y}$

By the product and chain rules,

$xz^2 + 2xyz \dfrac{\partial z}{\partial y} = 10y + 1$

Solve for the partial derivative, then evaluate at $(x,y,z) = (1,1,1)$.

$\dfrac{\partial z}{\partial y} = \dfrac{10y + 1 - xz^2}{2xyz}$

$\dfrac{\partial z}{\partial y} \bigg|_{x=1,y=1,z=1} = \dfrac{10 + 1 - 1}{2} = \boxed{5}$