let Z be open Subset of REAL; ( Z c= dom ((tan + cot) / exp_R) implies ( (tan + cot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x) ) ) )
A1:
for x being Real st x in Z holds
exp_R . x <> 0
by SIN_COS:54;
assume
Z c= dom ((tan + cot) / exp_R)
; ( (tan + cot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x) ) )
then
Z c= (dom (tan + cot)) /\ ((dom exp_R) \ (exp_R " {0}))
by RFUNCT_1:def 1;
then A2:
Z c= dom (tan + cot)
by XBOOLE_1:18;
then A3:
tan + cot is_differentiable_on Z
by Th6;
A4:
exp_R is_differentiable_on Z
by FDIFF_1:26, TAYLOR_1:16;
then A5:
(tan + cot) / exp_R is_differentiable_on Z
by A3, A1, FDIFF_2:21;
for x being Real st x in Z holds
(((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x)
proof
let x be
Real;
( x in Z implies (((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x) )
A6:
exp_R is_differentiable_in x
by SIN_COS:65;
A7:
exp_R . x <> 0
by SIN_COS:54;
assume A8:
x in Z
;
(((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x)
then A9:
(tan + cot) . x = (tan . x) + (cot . x)
by A2, VALUED_1:def 1;
tan + cot is_differentiable_in x
by A3, A8, FDIFF_1:9;
then diff (
((tan + cot) / exp_R),
x) =
(((diff ((tan + cot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((tan + cot) . x))) / ((exp_R . x) ^2)
by A6, A7, FDIFF_2:14
.=
(((((tan + cot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((tan + cot) . x))) / ((exp_R . x) ^2)
by A3, A8, FDIFF_1:def 7
.=
((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((tan + cot) . x))) / ((exp_R . x) ^2)
by A2, A8, Th6
.=
((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) * (exp_R . x)) - ((exp_R . x) * ((tan . x) + (cot . x)))) / ((exp_R . x) * (exp_R . x))
by A9, SIN_COS:65
.=
(((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - ((tan . x) + (cot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))
.=
(((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - ((tan . x) + (cot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x))
by XCMPLX_1:78
.=
(((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - ((tan . x) + (cot . x))) * (1 / (exp_R . x))
by A7, XCMPLX_1:60
.=
(((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - ((tan . x) + (cot . x))) / (exp_R . x)
;
hence
(((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x)
by A5, A8, FDIFF_1:def 7;
verum
end;
hence
( (tan + cot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((tan + cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) - (1 / ((sin . x) ^2))) - (tan . x)) - (cot . x)) / (exp_R . x) ) )
by A3, A4, A1, FDIFF_2:21; verum