let r be Real; :: thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ & Z c= dom (r (#) arccos ) holds
( r (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 )))) ) )
let Z be open Subset of REAL ; :: thesis: ( Z c= ].(- 1),1.[ & Z c= dom (r (#) arccos ) implies ( r (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 )))) ) ) )
assume that
A1: Z c= ].(- 1),1.[ and
A2: Z c= dom (r (#) arccos ) ; :: thesis: ( r (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 )))) ) )
A3: arccos is_differentiable_on Z by A1, FDIFF_1:34, SIN_COS6:108;
for x being Real st x in Z holds
((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 )))) )
assume A4: x in Z ; :: thesis: ((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 ))))
then A5: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
((r (#) arccos ) `| Z) . x = r * (diff arccos ,x) by A2, A3, A4, FDIFF_1:28
.= r * (- (1 / (sqrt (1 - (x ^2 ))))) by A5, SIN_COS6:108
.= - (r * (1 / (sqrt (1 - (x ^2 )))))
.= - (r / (sqrt (1 - (x ^2 )))) by XCMPLX_1:100 ;
hence ((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 )))) ; :: thesis: verum

end;

hence ( r (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arccos ) `| Z) . x = - (r / (sqrt (1 - (x ^2 )))) ) ) by A2, A3, FDIFF_1:28; :: thesis: verum