1. I think that if you put yourself back in the position of someone discovering this for the first time, the equality (under suitable hypotheses) $${\partial^2f\over\partial x\partial y}={\partial^2 f\over\partial y\partial x}\quad (1)$$ should count.

2. Here's a surprising application of that suprising equality. Suppose you're a profit-maximizing competitive firm, hiring both labor ($L$) (at a wage rate of $W$) and capital ($K$) (at a rental rate of $R$). Then an increase in $W$ will, in general, lead you to reduce your output and so employ less capital, but at the same time lead you to substitute capital for labor and so employ more capital. On balance, the derivative $dK/dW$ could be either positive or negative. Likewise for the derivative $dL/dR$. It does not seem to me to be at all intuitively obvious that these derivatives even have the same sign, much less that they are equal. But if one takes $f$ in (1) to be profit as a function of $x$ (labor) and $y$ (capital) then one discovers that in fact

$${dK\over dW}={dL\over dR}$$

(Of course this looks more symmetric if you write $X_1$ and $X_2$ for capital and labor, and $P_1$ and $P_2$ for the wage rate and the rental rate.)

1. I think that if you put yourself back in the position of someone discovering this for the first time, the equality (under suitable hypotheses) $${\partial^2f\over\partial x\partial y}={\partial^2 f\over\partial y\partial x}\quad (1)$$ should count.

2. Here's a surprising application of that suprising equality. Suppose you're a profit-maximizing competitive firm, hiring both labor ($L$) (at a wage rate of $W$) and capital ($K$) (at a rental rate of $R$). Then an increase in $W$ will, in general, lead you to reduce your output and so employ less capital, but at the same time lead you to substitute capital for labor and so employ more capital. On balance, the derivative $dK/dW$ could be either positive or negative. Likewise for the derivative $dL/dR$. It does not seem to me to be at all intuitively obvious that these derivatives even have the same sign, much less that they are equal. But if one takes $f$ in (1) to be profit as a function of $x$ (labor) and $y$ (capital) then one discovers that in fact

$${dK\over dW}={dL\over dR}$$

(Of course this looks more symmetric if you write $X_1$ and $X_2$ for labor and capital, and $P_1$ and $P_2$ for the wage rate and the rental rate.)