Find (∂/∂x)z at the point (x, y, z) = (0, 1, −1) if 2xy + z^2 cos(x) + zy = 8.

Question

Find (∂/∂x)z at the point (x, y, z) = (0, 1, −1) if 2xy + z² cos(x) + zy = 8.

Answer 1

Here z(x, y) is implicit function of 2 independent variables x and y. Let differentiate given equation partially by x: ∂/∂x(2xy + z² cos(x) + zy) = ∂/∂x(8), 2y + 2zcosx∂z/∂x - z²sinx + y∂z/∂x = 0;

From here ∂z/∂x = (z²sinx - 2y)/(2zcosx + y).

At the point (x, y, z) = (0, 1, -1) ∂z/∂x = ((-1)²sin0 - 2·1)/(2·(-1)cos0 + 1) = -2/(-1) = 2