What even are derivatives of ordinary differential equations (ODEs)? You might be confused. Isn't an ODE defined as \(u' = f'(u,t)\)? Thus, the derivative of any ODE is trivially \(f\)! Well, that is correct, but what if we have parameters \(p\) on our ODEs, so we have \(u' = f'(u,p,t)\), and we concern about the sensitivity of the parameters on the ODE solution? Let's find out!

Speaking of derivatives and the sensitivity analysis, let's review what are derivatives of a multi-variable function. When we have a function that is \(f: \mathbf{R}^n\mapsto \mathbf{R}^m\), we can then take a Jacobian matrix. When we have a function that is \(f: \mathbf{R}^n\mapsto \mathbf{R}\), then we can take the gradient vector. Analogously, we can take the Jacobian or the gradient of an ODE, which are called the forward sensitivity analysis and the adjoint sensitivity analysis respectively.

## Forward Sensitivity Analysis

Generally speaking, \(p\) and \(u\) are vectors, so \(\frac{\partial u}{\partial p}\) is a Jacobian matrix. We have

\[ \frac{d}{dt}\frac{\partial u}{\partial p} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial p} + \frac{\partial f}{\partial p}. \]

## Adjoint Sensitivity Analysis

To be continued...