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_1:9, SIN_COS6:106;
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_1:9;
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_1:9, SIN_COS6:106;
(((#Z n) * arccos) `| Z) . x = diff (((#Z n) * arccos),x) by A4, A5, FDIFF_1:def 7
.= (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_COS6:106
.= - ((n * ((arccos . x) #Z (n - 1))) * (1 / (sqrt (1 - (x ^2)))))
.= - ((n * ((arccos . x) #Z (n - 1))) / (sqrt (1 - (x ^2)))) by XCMPLX_1:99 ;
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_1:9; :: thesis: verum