Derivative of $a^T X^{-1} b$ with respect to $X$

Question

Given $a^T X^{-1} b$, where $a,b$ are $n\times 1$ vector, $X$ is $n \times n$ matrix. It is easy to see that $\frac{\partial a^{T} X b}{\partial X}=ab^{T}$, but what is $\frac{\partial a^{T} X^{-1} b}{\partial X}$?
If I do this $\frac{\partial a^{T} {X^{-1}} b}{\partial X}= \frac{\partial a^T X^{-1} b }{\partial X^{-1}} * \frac{\partial X^{-1} }{\partial X}$, what is $\frac{\partial X^{-1} }{\partial X}$?
Any hint?