¿Por qué necesito usar la matriz jacobiana?

In what context? The Jacobian is the first derivative of a function with respect to its variables in matrix form. Usually we have a Jacobian considered in terms of a function from RnRnRn to RmRmRm when we define it from the original analysis (calculus) definition. In statistical work (or machine learning, if you prefer), we often think of nnn being the sample size, but the Jacobian then would be from a ppp dimensional space (where ppp is the number of variables, so in, eg, regression, we'd include the intercept) and most often there is only one output. Edit: this is not the most interesting way you look at this… and technically we call the vector the gradient. See edit at the end for a more interesting example.

The Jacobian is incredibly useful in all sorts of mathematical problems, where a lot of statistics and machine learning get their theory from.

If you would care to be more explicit, and state the context in updating the question, I will change this answer if required. If you would like me to, please comment to this answer, so I know to do so.

Edit:

Based on the reply from the person who asked me to answer this question, we’d like some intuition on the Jacobian as it fits into ML and statistics. Most ML algorithms that use the Jacobian are statistically-based.

So we’re trying to learn a function fff from n data points to predict something. Say that our functional form has p parameters (for instance, some sort of regression.) However, instead of treating the functions output as our target, we’re going to look at the parameters as functions of the data, all crammed into one vector-valued function. We’ll call this function ggg.

Then our function g:Rn→Rpg:Rn→Rpg:Rn→Rp has a Jacobian defined by ∇g∇g∇g, which has element i, j as the first derivative of the parameter i with respect to variable j. Obviously linear regression wouldn’t be interesting here, but non-linear regression might very well be: we can see how each parameter changes with respect to each variable. That can be used in a lot of different ways, although it can be quite complex to do so well.