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, The Ramanujan Journal 13 (2007) pp. 167–201. In particular, his formula (1.3) is $$\frac{d^n\ }{dx^n} \phi^n(x) u(x) v(x) = \sum_{k=0}^n \binom nk \left(\frac{d^k\ }{dx^k} \phi^k(x) u(x)\right) \left(\frac{d^{n-k-1}}{dx^{n-k-1}}\phi^{n-k}(x) v'(x)\right)\tag{1}$$ 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 $(1)$ are discussed in my survey paper Lagrange Inversion, 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 https://terrytao.wordpress.com/2016/10/23/another-problem-about-power-series.