let Z be open Subset of REAL ; :: thesis:
assume that
A1: Z c= dom ((id Z) (#) arccos ) and
A2: Z c= ].(- 1),1.[ ; :: thesis:
A3: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
Z c= (dom (id Z)) /\ (dom arccos ) by A1, VALUED_1:def 4;
then A4: Z c= dom (id Z) by XBOOLE_1:18;
then A5: id Z is_differentiable_on Z by A3, FDIFF_1:31;
A6: arccos is_differentiable_on Z by A2, FDIFF_1:34, SIN_COS6:108;
for x being Real st x in Z holds
(((id Z) (#) arccos ) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2 ))))

proof

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
then A8: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
(((id Z) (#) arccos ) `| Z) . x = ((arccos . x) * (diff (id Z),x)) + (((id Z) . x) * (diff arccos ,x)) by A1, A5, A6, A7, FDIFF_1:29
.= ((arccos . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff arccos ,x)) by A5, A7, FDIFF_1:def 8
.= ((arccos . x) * 1) + (((id Z) . x) * (diff arccos ,x)) by A4, A3, A7, FDIFF_1:31
.= ((arccos . x) * 1) + (((id Z) . x) * (- (1 / (sqrt (1 - (x ^2 )))))) by A8, SIN_COS6:108
.= (arccos . x) + (x * (- (1 / (sqrt (1 - (x ^2 )))))) by A7, FUNCT_1:35
.= (arccos . x) - (x * (1 / (sqrt (1 - (x ^2 )))))
.= (arccos . x) - (x / (sqrt (1 - (x ^2 )))) by XCMPLX_1:100 ;
hence (((id Z) (#) arccos ) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2 )))) ; :: thesis:

end;

hence ( (id Z) (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccos ) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2 )))) ) ) by A1, A5, A6, FDIFF_1:29; :: thesis: