let Z be open Subset of REAL; ( Z c= dom (cot * exp_R) implies ( - (cot * exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2) ) ) )
assume A1:
Z c= dom (cot * exp_R)
; ( - (cot * exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2) ) )
then A2:
Z c= dom (- (cot * exp_R))
by VALUED_1:8;
A3:
cot * exp_R is_differentiable_on Z
by A1, FDIFF_8:13;
then A4:
(- 1) (#) (cot * exp_R) is_differentiable_on Z
by A2, FDIFF_1:20;
A5:
for x being Real st x in Z holds
sin . (exp_R . x) <> 0
for x being Real st x in Z holds
((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2)
proof
let x be
Real;
( x in Z implies ((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2) )
assume A6:
x in Z
;
((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2)
A7:
exp_R is_differentiable_in x
by SIN_COS:65;
A8:
sin . (exp_R . x) <> 0
by A5, A6;
then A9:
cot is_differentiable_in exp_R . x
by FDIFF_7:47;
A10:
cot * exp_R is_differentiable_in x
by A3, A6, FDIFF_1:9;
((- (cot * exp_R)) `| Z) . x =
diff (
(- (cot * exp_R)),
x)
by A4, A6, FDIFF_1:def 7
.=
(- 1) * (diff ((cot * exp_R),x))
by A10, FDIFF_1:15
.=
(- 1) * ((diff (cot,(exp_R . x))) * (diff (exp_R,x)))
by A7, A9, FDIFF_2:13
.=
(- 1) * ((- (1 / ((sin . (exp_R . x)) ^2))) * (diff (exp_R,x)))
by A8, FDIFF_7:47
.=
(- 1) * (- ((diff (exp_R,x)) / ((sin . (exp_R . x)) ^2)))
.=
(exp_R . x) / ((sin . (exp_R . x)) ^2)
by SIN_COS:65
;
hence
((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2)
;
verum
end;
hence
( - (cot * exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cot * exp_R)) `| Z) . x = (exp_R . x) / ((sin . (exp_R . x)) ^2) ) )
by A4; verum