Final answer:
To find the partial derivative dz(1,0), we need to find the partial derivatives of z with respect to x and y. After differentiating the equation with respect to x and y and simplifying, we can substitute x = 1 and y = 0 to calculate dz(1,0).
Step-by-step explanation:
To find the partial derivative dz(1,0), we first need to find the partial derivatives of z with respect to x and y.
Given the equation x + y + xz + yz² - 1 = 0, we differentiate both sides with respect to x to get 1 + z + x * dz/dx + y * dz/dx + yz² * dz/dx = 0.
Simplifying, we have dz/dx = -1 / (1 + z + 2yxz).
Next, we differentiate both sides with respect to y to get 1 + z + x * dz/dy + y * dz/dy + 2yz * dz/dy = 0.
Simplifying, we have dz/dy = -1 / (1 + z + 2xyz).
Finally, substituting x = 1 and y = 0 into the expressions for dz/dx and dz/dy, we can calculate dz(1,0). It is recommended to use a calculator or software to perform the computations.