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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​

15. The equation below defines z as a differentiable function of x and y. Find the-example-1

1 Answer

3 votes

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}

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