Section 14.5 Chain Rule

A manufacturer has modeled its yearly production function \(P\) (the value of its entire production, in millions of dollars) as a Cobb-Douglas function \(P(L, K) = 1.47L^{0.65}K^{0.35}\) where \(L\) is the number of labor hours (in thousands) and \(K\) is the invested capital (in millions of dollars). Suppose that when \(L = 30\) and \(K = 8\), the labor force is decreasing at a rate of 2000 labor hours per year and capital is increasing at a rate of $500,000 per year. Find the rate of change of production.