Note that the following sub-questions are not related to each other. (1) Suppose that y is a function of x satisfying the equation 5x² + 10y + 3xy + 5y² + 6x = 0. Use the implicit differentiation rule to find dy/dx (show the necessary steps).
(2) Assume that we have a Cobb-Douglas production function for a certain industry: Q=cLᵅ Kᵝ, where c is a constant, and L and K represent labor input and capital input, respectively. Assume that the production is fixed at Q = Qo, and at this fixed level of production, K is a function of L defined by Qo=cLᵅ Kᵝ. Use the implicit differentiation rule to find the "rate of substitution of capital for labor", which is measured by aK/aL when the production is fixed at Q = Qo (show the necessary steps). (3) For x > 0 and y > 0, the equation y³/x³ - (x + 2)²(y + 3) = 0 defines y as an implicit function of x. Find the elasticity of y with respect to x at the point (xo, yo), where хо and Yo are some known values.