Determine the Jacobian matrix for $f(x) = \sqrt{x^2+1}-1$
I'm confused because we only have one unknown variable. Usually you have two or more unknowns, something like $f(x,y,z)=..$ mixed function with several variables. Then you determine its Jacobian matrix.
But in this case we only have one variable. So what I would do is actually the same as usual, derivate for $x$ and then that would be our Jacobian matrix?
$$\frac{\vartheta }{\vartheta x}f(x) = \frac{x}{\sqrt{x^2+1}}$$
$$\Rightarrow \text{Jacobian matrix } J_f = \begin{pmatrix} \frac{x}{\sqrt{x^2+1}} \end{pmatrix}$$
Really...?
For a function of one variable the Jacobian reduce to the first derivative, that is: $$ J_f=\frac{df}{dx}=\frac{x}{\sqrt{x^2+1}} $$