We have a function, say, a profit function of a monopoly firm $$Y=R(Q)-C(Q,b)$$ where:
$R$ is the revenue function;
$Q$ is the output quantity
$C$ is the cost function
$b$ is the price of an input (in this particular problem it's oil)
We need to determine the sign of $\frac {dQ}{db}$.
One clue given is that: $$\frac {\partial^2 C}{\partial b \partial Q} > 0$$ which means that the marginal cost of producing one more unit increases when the input price increases.
However, as comparative statics suggest, the thing we need is the $\frac {\partial^2 Y}{\partial Q \partial b}$, that actually implies finding the sign of $-\frac {\partial^2 C}{\partial Q \partial b}$.
The question is, if we know the sign of one partial second erivative $\frac {\partial^2 C}{\partial b \partial Q} > 0$, is there any rule that helps us find the sign of $\frac {\partial^2 C}{\partial Q \partial b}$ (the "vice-versa" partial second derivative) in a straightforward way?