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_1:26, SIN_COS6:106;
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_1:20
.= r * (- (1 / (sqrt (1 - (x ^2))))) by A5, SIN_COS6:106
.= - (r * (1 / (sqrt (1 - (x ^2)))))
.= - (r / (sqrt (1 - (x ^2)))) by XCMPLX_1:99 ;
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_1:20; :: thesis: verum