Applications of Partial Derivatives

Find the marginal productivities of capital (K) and labour (L) if P = 8L – 2K + 3K² – 2L² + 7KL when K = 3 and L = 1.

If the production of a firm is given by P = 4LK – L² + K², L > 0, K > 0, Prove that L `(del"P")/(del"L") + "K"(del"P")/(del"K")` = 2P.

If the production function is z = 3x² – 4xy + 3y² where x is the labour and y is the capital, find the marginal productivities of x and y when x = 1, y = 2.

For the production function P = 3(L)^0.4 (K)^0.6, find the marginal productivities of labour (L) and capital (K) when L = 10 and K = 6. [use: (0.6)^0.6 = 0.736, (1.67)^0.4 = 1.2267]

The demand for a quantity A is q = `13 - 2"p"_1 - 3"p"_2^2`. Find the partial elasticities `"E"_"q"/("E"_("p"_1))` and `"E"_"q"/("E"_("p"_2))`  when p₁ = p₂ = 2.

The demand for a quantity A is q = `80 - "p"_1^2 + 5"p"_2 - "p"_1"p"_2`. Find the partial elasticities `"E"_"q"/("E"_("p"_1))` and `"E"_"q"/("E"_("p"_2))`  when p₁ = 2, p₂ = 1.

If f(x, y) = 3x² + 4y³ + 6xy - x²y³ + 7, then show that f_yy (1,1) = 18.

Verify `(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)` for u = x³ + 3x² y² + y³.