A detailed historical discussion of identities like this one can be found in Warren P. Johnson's paper *[The Pfaff/Cauchy derivative identities and Hurwitz type extensions](https://doi.org/10.1007/s11139-006-0246-0)*, The Ramanujan Journal 13 (2007) pp. 167–201.
 In particular, his formula (1.3) is
$$\frac{d^n\ }{dx^n} \phi^n(x) u(x) v(x) = \sum_{k=0}^n \binom nk \left(\frac{d^k\ }{dx^k} \phi^k(x) u(x)\right)
 \left(\frac{d^{n-k-1}}{dx^{n-k-1}}\phi^{n-k}(x) v'(x)\right)\tag{1}\label{1}$$
where for $n=k$, $\frac{d^{n-k-1}}{dx^{n-k-1}}\phi^{n-k}(x) v'(x)$ is to be interpreted as $v(x)$. This identity was given by Cauchy in 1826, but is equivalent to an identity given by Pfaff in 1795.

The OP's identity is equivalent to the case $\phi(x)=f$, $u(x)=1$, $v(x) = x$. 

There is a related formula in Cayley's paper *On the partitions of a polygon*, Coll. Math. Papers 13 (1897), 93–113; Proc. London Math. Soc. (1) 22 (1891), 237–262, but I didn't see this formula there. 

Formulas like \eqref{1} are discussed in 
my survey paper *[Lagrange Inversion][2]*, Journal of Combinatorial Theory, Series A 144 (2016), pp. 212–249, section 2.6.

There is also a discussion of a somewhat related formula on Terry Tao's blog at [Another problem about power series](https://terrytao.wordpress.com/2016/10/23/another-problem-about-power-series).

*Added November 12, 2022:* A new paper with multivariable generalizations of these identities is Wenchang Chu, [Multiple derivative inversions and Lagrange-Good expansion formulae][3] ([DOI](https://doi.org/10.3390/math10224234) is being registered), Mathematics 10 (2022), 4234. 


  [1]: http://link.springer.com/article/10.1007/s11139-006-0246-0
  [2]: https://arxiv.org/abs/1609.05988
  [3]: https://www.mdpi.com/2227-7390/10/22/4234