Differentiation of unit force vector

Question

I was reading a paper and don't know how the following was derived. Given that $f = \begin{bmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} + g \\ \end{bmatrix}$ and $v = \begin{bmatrix} \dddot{x} \\ \dddot{y} \\ \dddot{z} \\ \end{bmatrix}$, I thought that $\frac{\dot{f}}{\lVert f\rVert}$ would be equal to $\frac{v}{\lVert f\rVert}$. However, the equation in the paper states that $\frac{\dot{f}}{\lVert f\rVert} = \frac{v}{\lVert f\rVert} - \frac{ff^Tv}{\lVert f\rVert^3}$. I don't understand where the second term comes from and am not sure where to start. Could someone help explain this? Thank you in advance!

Answer 1

I've not done the math, but that looks like it is coming from the total differential of $f$. I suppose $f$ is a force vector of a moving body? Some more context information would be useful.