# Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented structure , and an output. In the way of studying the sensitivity analysis of such models, I think of using local sensitivity analysis tests, in which the partial derivative of the models is computed to represent the index of sensitivity of each parameter or variable. My question here is that, If my model cannot be represented by a function to evluate partial derivatives explicitly, is it possible to use directly the numerical methods to derive the partial derivatives? For example, we may run the model for some values $x_1, x_2, ...,x_j,..., x_n$ to give $y_j$, then for the values $x_1, x_2, ...,x_j+\Delta_j,... , x_n$ to give $y_{j+\Delta}$, and then estimate the partial derivative wrt the variable $x_j$ by $\frac{y_{j+\Delta}-y_j}{\Delta}$. I think this way is straightforward, but how to be assure that our model has a differentiable form i.e. is differentiable? Or let me say, can we speak about difference in variation without including the notion of partial derivatives and differentiability of the model?

I respect any perspective and I highly appriciate your opinions.

Thanks .

The answer is yes, and no, and maybe. In sensitivity analysis, we can always run a model at different parameter values and use finite differencing methods to compute the directional derivatives in our parameter space. This is, as you might imagine, of limited use without any other external information. First, it requires many runs. Second, it only gives us scraps of information. For instance, computing the change with respect to some parameter $\theta$ and separately computing the change with respect to some other parameter $\gamma$ doesn't tells us anything about whether the parameters are somehow dependent, or what happens with convex combinations of parameters, etc. If we know something about the model, then we might be able to extrapolate these data. But in a vacuum, no.