`f_n(x)=e^(f_(n-1)(x))` for all `n in Na n df_0(x)=x ,t h e n d/(dx){f_n(x)}` is `(f_n(x)d)/(dx){f_(n-1)(x)}` (b) `f_n(x)f_(n-1)(x)` `f_n(x)f_(n-1)(x)f_2(x)dotf_1(x)` `non eoft h e s e`

Options

A. `f_(n)(x)(d)/(dx){f_(n-1)(x)}`
B. `f_(n)(x)f_(n-1)(x)`
C. `f_(n)(x)f_(n-1)(x)...f_(2)(x).f_(1)(x)`
D. None of these

Explanation

`(d)/(dx){f_(n)(x)}=(d)/(dx){e^(f_(n-1)(x))}`
`=e^(f_(n-1)(x))(d)/(dx){f_(n-1)(x)}=f_(n)(x)(d)/(dx){f_(n-1)(x)}`
`f_(n)(x).(d)/(dx){e^(f_(n-2)(x))}=f_(n)(x).e^(f_(n-2)(x))(d)/(dx){f_(n-2)(x)}`
`=f_(n)(x)f_(n-1)(x)(d)/(dx){f_(n-2)(x)}` ...
`=f_(n)(x)f_(n-1)(x)...f_(2)(x)(d)/(dx){f_(1)(x)}`
`=f_(n)(x)f_(n-1)(x)...f_(2)(x)(d)/(dx){e^(f_(0)(x))}`
`=f_(n)(x)f_(n-1)(x)...f_(2)(x)e^(f_(0)(x))(d)/(dx){f_(0)(x)}`
`"Use "e^(f_(0)(x))=f_(1)(x)and f_(0)(x)=x`