Question

Assume that F(x,y,z(x,y))=0 implicitly defines z as a differentiable function of x and y. The partial derivatives of z are ∂x∂z=−Fz/Fx and ∂y/∂z=−Fz/Fy. Evaluate ∂x/∂z and ∂y/∂z for xyz+8x+9y−7z=0. ∂x/∂z=

Answer 1

For the implicit function xyz + 8x + 9y - 7z = 0, the partial derivatives of z with respect to x and y are evaluated using the expressions for Fx, Fy, and Fz, which are yz + 8, xz + 9, and xy - 7, respectively.

Step-by-step explanation:

To evaluate partial derivatives of z with respect to x and y for the given implicit function xyz + 8x + 9y - 7z = 0, we must first calculate Fx, Fy, and Fz, which represent partial derivatives of the function F with respect to x, y, and z respectively.

The correct expressions for these partial derivatives are:

• Fx = yz + 8
• Fy = xz + 9
• Fz = xy - 7

Now, according to the given information, partial derivatives of z can be found using the formulas:

• ∂x/∂z = -Fz/Fx
• ∂y/∂z = -Fz/Fy

Substitute the values of Fx, Fy, and Fz into these formulas to obtain:

• ∂x/∂z = -(xy - 7) / (yz + 8)
• ∂y/∂z = -(xy - 7) / (xz + 9)