let Z be open Subset of REAL; ( 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.[
; ( 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
A4:
for x being Real st x in Z holds
sec * arccot is_differentiable_in x
then A7:
sec * arccot is_differentiable_on Z
by A1, FDIFF_1:9;
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;
( x in Z implies ((sec * arccot) `| Z) . x = - ((sin . (arccot . x)) / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) )
assume A8:
x in Z
;
((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_1:9;
((sec * arccot) `| Z) . x =
diff (
(sec * arccot),
x)
by A7, A8, FDIFF_1:def 7
.=
(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 7
.=
((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_1:76
.=
- ((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))))
;
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_1:9; verum