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

proof

let x be Real; :: thesis: ( x in Z implies (#Z n) * arccos is_differentiable_in x )
assume x in Z ; :: thesis: (#Z n) * arccos is_differentiable_in x
then arccos is_differentiable_in x by A2, FDIFF_1:16, SIN_COS6:108;
hence (#Z n) * arccos is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A4: (#Z n) * arccos is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
(((#Z n) * arccos ) `| Z) . x = - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies (((#Z n) * arccos ) `| Z) . x = - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2 )))) )
assume A5: x in Z ; :: thesis: (((#Z n) * arccos ) `| Z) . x = - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2 ))))
then A6: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
A7: arccos is_differentiable_in x by A2, A5, FDIFF_1:16, SIN_COS6:108;
(((#Z n) * arccos ) `| Z) . x = diff ((#Z n) * arccos ),x by A4, A5, FDIFF_1:def 8
.= (n * ((arccos . x) #Z (n - 1))) * (diff arccos ,x) by A7, TAYLOR_1:3
.= (n * ((arccos . x) #Z (n - 1))) * (- (1 / (sqrt (1 - (x ^2 ))))) by A6, SIN_COS6:108
.= - ((n * ((arccos . x) #Z (n - 1))) * (1 / (sqrt (1 - (x ^2 )))))
.= - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2 )))) by XCMPLX_1:100 ;
hence (((#Z n) * arccos ) `| Z) . x = - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2 )))) ; :: thesis: verum

end;

hence ( (#Z n) * arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arccos ) `| Z) . x = - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2 )))) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum