let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sec * arccot ) & Z c= ].(- 1),1.[ implies ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) ) ) )
assume that
A1: Z c= dom (sec * arccot ) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) ) )
A3: for x being Real st x in Z holds
cos . (arccot . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (arccot . x) <> 0 )
assume x in Z ; :: thesis: cos . (arccot . x) <> 0
then arccot . x in dom sec by A1, FUNCT_1:21;
hence cos . (arccot . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

A4: for x being Real st x in Z holds
sec * arccot is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec * arccot is_differentiable_in x )
assume A5: x in Z ; :: thesis: sec * arccot is_differentiable_in x
then cos . (arccot . x) <> 0 by A3;
then A6: sec is_differentiable_in arccot . x by FDIFF_9:1;
arccot is_differentiable_on Z by A2, SIN_COS9:82;
then arccot is_differentiable_in x by A5, FDIFF_1:16;
hence sec * arccot is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: sec * arccot is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) )
assume A8: x in Z ; :: thesis: ((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 ))))
then A9: cos . (arccot . x) <> 0 by A3;
cos . (arccot . x) <> 0 by A3, A8;
then A10: sec is_differentiable_in arccot . x by FDIFF_9:1;
A11: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A12: arccot is_differentiable_in x by A8, FDIFF_1:16;
((sec * arccot ) `| Z) . x = diff (sec * arccot ),x by A7, A8, FDIFF_1:def 8
.= (diff sec ,(arccot . x)) * (diff arccot ,x) by A12, A10, FDIFF_2:13
.= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2 )) * (diff arccot ,x) by A9, FDIFF_9:1
.= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2 )) * ((arccot `| Z) . x) by A8, A11, FDIFF_1:def 8
.= ((sin . (arccot . x)) / ((cos . (arccot . x)) ^2 )) * (- (1 / (1 + (x ^2 )))) by A2, A8, SIN_COS9:82
.= - (((sin . (arccot . x)) / ((cos . (arccot . x)) ^2 )) * (1 / (1 + (x ^2 ))))
.= - (((sin . (arccot . x)) * 1) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) by XCMPLX_1:77
.= - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) ;
hence ((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) ; :: thesis: verum

end;

hence ( sec * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * arccot ) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2 ) * (1 + (x ^2 )))) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum