let x be Real; :: thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x & f . x > - 1 & f . x < 1 holds
( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_in x & f . x > - 1 & f . x < 1 implies ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) )
assume that
A1: f is_differentiable_in x and
A2: f . x > - 1 and
A3: f . x < 1 ; :: thesis: ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) )
f . x in ].(- 1),1.[ by A2, A3, XXREAL_1:4;
then A4: arctan is_differentiable_in f . x by Th73, FDIFF_1:9;
then diff ((arctan * f),x) = (diff (arctan,(f . x))) * (diff (f,x)) by A1, FDIFF_2:13
.= (diff (f,x)) * (1 / (1 + ((f . x) ^2))) by A2, A3, Th75
.= (diff (f,x)) / (1 + ((f . x) ^2)) ;
hence ( arctan * f is_differentiable_in x & diff ((arctan * f),x) = (diff (f,x)) / (1 + ((f . x) ^2)) ) by A1, A4, FDIFF_2:13; :: thesis: verum