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).

x² - 5y² + xyz² = y - 4​

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`.

`(\partial(xyz^2))/(\partial y) = (\partial(-x^2 + 5y^2 + y - 4))/(\partial y)`

By the product and chain rules,

`xz^2 + 2xyz (\partial z)/(\partial y) = 10y + 1`

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

`(\partial z)/(\partial y) = (10y + 1 - xz^2)/(2xyz)`

`(\partial z)/(\partial y) \bigg|_(x=1,y=1,z=1) = (10 + 1 - 1)/(2) = \boxed{5}`