let Z be open Subset of REAL; ( Z c= dom (tan * arccot) & Z c= ].(- 1),1.[ implies ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1:
Z c= dom (tan * arccot)
and
A2:
Z c= ].(- 1),1.[
; ( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) )
A3:
for x being Real st x in Z holds
tan * arccot is_differentiable_in x
then A6:
tan * arccot is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
proof
let x be
Real;
( x in Z implies ((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) )
assume A7:
x in Z
;
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
then
arccot . x in dom tan
by A1, FUNCT_1:11;
then A8:
cos . (arccot . x) <> 0
by FDIFF_8:1;
then A9:
tan is_differentiable_in arccot . x
by FDIFF_7:46;
A10:
arccot is_differentiable_on Z
by A2, SIN_COS9:82;
then A11:
arccot is_differentiable_in x
by A7, FDIFF_1:9;
((tan * arccot) `| Z) . x =
diff (
(tan * arccot),
x)
by A6, A7, FDIFF_1:def 7
.=
(diff (tan,(arccot . x))) * (diff (arccot,x))
by A11, A9, FDIFF_2:13
.=
(1 / ((cos . (arccot . x)) ^2)) * (diff (arccot,x))
by A8, FDIFF_7:46
.=
(1 / ((cos . (arccot . x)) ^2)) * ((arccot `| Z) . x)
by A7, A10, FDIFF_1:def 7
.=
(1 / ((cos . (arccot . x)) ^2)) * (- (1 / (1 + (x ^2))))
by A2, A7, SIN_COS9:82
.=
- ((1 / ((cos . (arccot . x)) ^2)) * (1 / (1 + (x ^2))))
.=
- (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
by XCMPLX_1:102
;
hence
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2))))
;
verum
end;
hence
( tan * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * arccot) `| Z) . x = - (1 / (((cos . (arccot . x)) ^2) * (1 + (x ^2)))) ) )
by A1, A3, FDIFF_1:9; verum