Final answer:
To find the total differential for the function z=2x²+1/2 xy-3y³, we use the formula dz = ∂z/∂x * dx + ∂z/∂y * dy. We find the partial derivatives ∂z/∂x and ∂z/∂y, and substitute them into the formula to find the total differential.
Step-by-step explanation:
To find the total differential for the function z=2x²+1/2 xy-3y³, we need to find dz which represents the change in z for a small change in x and y. The total differential can be found using the formula dz = ∂z/∂x * dx + ∂z/∂y * dy, where ∂z/∂x and ∂z/∂y are the partial derivatives of z with respect to x and y respectively.
First, we find the partial derivative ∂z/∂x:
∂z/∂x = 4x + (1/2)y
Next, we find the partial derivative ∂z/∂y:
∂z/∂y = (1/2)x - 9y²
Finally, we substitute these values into the formula dz = ∂z/∂x * dx + ∂z/∂y * dy to find the total differential for the function z=2x²+1/2 xy-3y³.