Kettenregel - mehrere Kompositionen
1. Satz
\begin{align*}
f_n(x) \coloneqq& u_1(u_2(...(u_n(x)))) \\
f_n(x)' =& u_1'(u_2(...(u_n(x)))) \cdot u_2'(...(u_n(x))) \cdot ... \cdot u_n'(x)
\end{align*}
2. Beweis
2.1. IA
- siehe: Kettenregel
2.2. IS
\begin{align*}
f_{n}'(x) =& u_1'(u_2(...(u_n(x)))) \cdot u_2'(...(u_n(x))) \cdot ... \cdot u_n'(x)\\
f_n(x) \coloneqq& u_1(u_2(...(u_n((x))))) \\
f_{n+1}(x) \coloneqq& u_1(u_2(...(u_n(u_{n+1}(x))))) \\
f_{n+1}(x) =& f_n(u_{n+1}(x))
\end{align*}
\begin{align*}
f_{n+1}(x)' =& f_n'(u_{n+1}(x)) \cdot u'_n(x) \\
=& \left[u_1'(u_2(...(u_n(u_{n+1}(x))))) \cdot u_2'(...(u_n(u_{n+1}(x)))) ... \cdot u_n'(u_{n+1}(x)) \right] \cdot u_n'(x) \\
f_{n+1}(x)' =& u_1'(u_2(...(u_{n+1}(x)))) \cdot u_2'(...(u_{n+1}(x))) \cdot ... \cdot u_{n+1}'(x)
\end{align*}