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
( arcsin * f is_differentiable_in x & diff ((arcsin * f),x) = (diff (f,x)) / (sqrt (1 - ((f . x) ^2))) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_in x & f . x > - 1 & f . x < 1 implies ( arcsin * f is_differentiable_in x & diff ((arcsin * f),x) = (diff (f,x)) / (sqrt (1 - ((f . x) ^2))) ) )
assume that
A1: f is_differentiable_in x and
A2: ( f . x > - 1 & f . x < 1 ) ; :: thesis: ( arcsin * f is_differentiable_in x & diff ((arcsin * f),x) = (diff (f,x)) / (sqrt (1 - ((f . x) ^2))) )
f . x in ].(- 1),1.[ by A2;
then A3: arcsin is_differentiable_in f . x by FDIFF_1:9, SIN_COS6:83;
then diff ((arcsin * f),x) = (diff (arcsin,(f . x))) * (diff (f,x)) by A1, FDIFF_2:13
.= (diff (f,x)) * (1 / (sqrt (1 - ((f . x) ^2)))) by A2, SIN_COS6:83
.= (diff (f,x)) / (sqrt (1 - ((f . x) ^2))) by XCMPLX_1:99 ;
hence ( arcsin * f is_differentiable_in x & diff ((arcsin * f),x) = (diff (f,x)) / (sqrt (1 - ((f . x) ^2))) ) by A1, A3, FDIFF_2:13; :: thesis: verum