Graduate Derivation of P.D.F. from distribution function

[WMDhamnekar]

TL;DR

If $F_{X_k}(x) = p(X \leq x) = \displaystyle\sum_{j=k}^n \binom{n}{j} F^j(x)(1-F(x))^{n-j}, -\infty < x < \infty$ then how to prove $f_{X_k} (x) =\frac{n!}{(k-1)!(n-k)!}f(x) F^{k-1}(x)(1-F(x))^{n-k}$

Author computed $f_{X_k}(x)$ as follows but I don't understand it. Would any member explain me the following computations?

 

[WMDhamnekar]

Summary: If $F_{X_k}(x) = p(X \leq x) = \displaystyle\sum_{j=k}^n \binom{n}{j} F^j(x)(1-F(x))^{n-j}, -\infty < x < \infty$ then how to prove $f_{X_k} (x) =\frac{n!}{(k-1)!(n-k)!}f(x) F^{k-1}(x)(1-F(x))^{n-k}$

Author computed $f_{X_k}(x)$ as follows but I don't understand it. Would any member explain me the following computations?

I tag this question as "SOLVED". I understood all the computations. Thanks.