Given z = f( x, y) = 3x² + xy + 4y², find the partial derivatives.
To find the partial derivatives of z with respect to x and y, we take the derivative of f(x, y) with respect to x and y separately.
∂z/∂x = ∂/∂x (3x² + xy + 4y²)
∂z/∂x = 6x + y
∂z/∂y = ∂/∂y (3x² + xy + 4y²)
∂z/∂y = x + 8y
Therefore, the partial derivatives of z with respect to x and y are:
∂z/∂x = 6x + y
∂z/∂y = x + 8y