:: Several Differentiable Formulas of Special Functions -- Part {II}
:: by Yan Zhang , Bo Li and Xiquan Liang
::
:: Copyright (c) 2005-2021 Association of Mizar Users

Lm1: for x being Real holds 1 - (cos (2 * x)) = 2 * ((sin x) ^2)
proof end;

Lm2: for x being Real holds 1 + (cos (2 * x)) = 2 * ((cos x) ^2)
proof end;

Lm3: for x being Real st sin . x > 0 holds
sin . x = ((1 - (cos . x)) * (1 + (cos . x))) #R (1 / 2)

proof end;

Lm4: for x being Real st sin . x > 0 & cos . x < 1 & cos . x > - 1 holds
(sin . x) / ((1 - (cos . x)) #R (1 / 2)) = (1 + (cos . x)) #R (1 / 2)

proof end;

:: a ^ x=exp.(x*ln a)
theorem Th1: :: FDIFF_6:1
for x, a being Real st a > 0 holds
exp_R . (x * (log (number_e,a))) = a #R x
proof end;

:: 15 a ^ (-x)=exp.(-x*ln a)
theorem Th2: :: FDIFF_6:2
for x, a being Real st a > 0 holds
exp_R . (- (x * (log (number_e,a)))) = a #R (- x)
proof end;

Lm5: for x being Real st sin . x > 0 & cos . x < 1 & cos . x > - 1 holds
(sin . x) / ((1 + (cos . x)) #R (1 / 2)) = (1 - (cos . x)) #R (1 / 2)

proof end;

:: 1 f.x=a^2-x^2
theorem Th3: :: FDIFF_6:3
for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 - f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 holds
( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) | Z) . x = - (2 * x) ) )
proof end;

:: 2 f.x=(a^2+x^2)/(a^2-x^2)
theorem Th4: :: FDIFF_6:4
for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x <> 0 ) ) holds
( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) | Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) )
proof end;

:: 3 f.x=ln((a^2+x^2)/(a^2-x^2))
theorem Th5: :: FDIFF_6:5
for a being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom f & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f | Z) . x = ((4 * (a ^2)) * x) / ((a |^ 4) - (x |^ 4)) ) )
proof end;

:: 4 f.x=(1/(4*a^2))*ln((a^2+x^2)/(a^2-x^2))
theorem :: FDIFF_6:6
for a being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds
( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) | Z) . x = x / ((a |^ 4) - (x |^ 4)) ) )
proof end;

:: 5 f.x=x^2/(1+x^2)
theorem Th7: :: FDIFF_6:7
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 ) ) holds
( f1 / (f2 + f1) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / (f2 + f1)) | Z) . x = (2 * x) / ((1 + (x ^2)) ^2) ) )
proof end;

:: 6 f.x=(1/2)*ln(x^2/(1+x^2))
theorem :: FDIFF_6:8
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) f) & f = ln * (f1 / (f2 + f1)) & f1 = #Z 2 & ( for x being Real st x in Z holds
( f2 . x = 1 & x <> 0 ) ) holds
( (1 / 2) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) f) | Z) . x = 1 / (x * (1 + (x ^2))) ) )
proof end;

:: 7 f.x=ln(x^n)
theorem :: FDIFF_6:9
for n being Element of NAT
for Z being open Subset of REAL st Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds
x > 0 ) holds
( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (#Z n)) | Z) . x = n / x ) )
proof end;

:: 8 f.x=1/x+ln((x-1)/x)
theorem :: FDIFF_6:10
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f2 ^) + (ln * (f1 / f2))) & ( for x being Real st x in Z holds
( f2 . x = x & f2 . x > 0 & f1 . x = x - 1 & f1 . x > 0 ) ) holds
( (f2 ^) + (ln * (f1 / f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f2 ^) + (ln * (f1 / f2))) | Z) . x = 1 / ((x ^2) * (x - 1)) ) )
proof end;

:: 10 f.x=a ^ x=exp.(x*ln a)
theorem Th11: :: FDIFF_6:11
for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom () & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 holds
( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
(() | Z) . x = (a #R x) * (log (number_e,a)) ) )
proof end;

:: 11 f.x=(a ^ x)*(x-1/ln a)/ln a
theorem :: FDIFF_6:12
for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds
( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds
( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) | Z) . x = x * (a #R x) ) )
proof end;

:: 12 f.x=a ^ x*exp(x)/(1+ln a)
theorem :: FDIFF_6:13
for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 + (log (number_e,a)))) (#) (() (#) exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e holds
( (1 / (1 + (log (number_e,a)))) (#) (() (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) (() (#) exp_R)) | Z) . x = (a #R x) * () ) )
proof end;

:: 13 f.x=exp_R(-x)
theorem Th14: :: FDIFF_6:14
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom () & ( for x being Real st x in Z holds
f . x = - x ) holds
( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
(() | Z) . x = - (exp_R (- x)) ) )
proof end;

:: 14 f.x=-(x+1)*exp_R(-x)
theorem :: FDIFF_6:15
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 (#) (exp_R * f2)) & ( for x being Real st x in Z holds
( f1 . x = (- x) - 1 & f2 . x = - x ) ) holds
( f1 (#) (exp_R * f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 (#) (exp_R * f2)) | Z) . x = x / () ) )
proof end;

:: 16 f.x=a ^ (-x)=exp_R.(-x*ln a)
theorem Th16: :: FDIFF_6:16
for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (- ()) & ( for x being Real st x in Z holds
f . x = - (x * (log (number_e,a))) ) & a > 0 holds
( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) | Z) . x = (a #R (- x)) * (log (number_e,a)) ) )
proof end;

:: 17 f.x=-(a ^ -x)*(x+1/ln a)/ln a
theorem :: FDIFF_6:17
for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) & ( for x being Real st x in Z holds
( f1 . x = - (x * (log (number_e,a))) & f2 . x = x + (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds
( (1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((- (exp_R * f1)) (#) f2)) | Z) . x = x / (a #R x) ) )
proof end;

:: 18 f.x=(1/(ln a-1))*a ^ x/exp_R(x)
theorem :: FDIFF_6:18
for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) (() / exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds
( (1 / ((log (number_e,a)) - 1)) (#) (() / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / ((log (number_e,a)) - 1)) (#) (() / exp_R)) | Z) . x = (a #R x) / () ) )
proof end;

:: 19 f.x=(1/(1-ln a))*exp_R(x)/a ^ x
theorem :: FDIFF_6:19
for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / ())) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds
( (1 / (1 - (log (number_e,a)))) (#) (exp_R / ()) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / ())) | Z) . x = () / (a #R x) ) )
proof end;

:: 20 f.x=ln(exp_R.x+1)
theorem :: FDIFF_6:20
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (ln * ()) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( ln * () is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * ()) | Z) . x = () / (() + 1) ) )
proof end;

:: 21 f.x=ln(exp_R.x-1)
theorem :: FDIFF_6:21
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (ln * ()) & ( for x being Real st x in Z holds
( f . x = 1 & () . x > 0 ) ) holds
( ln * () is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * ()) | Z) . x = () / (() - 1) ) )
proof end;

:: 22 f.x=-ln(1-exp_R.x)
theorem :: FDIFF_6:22
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (- (ln * ())) & ( for x being Real st x in Z holds
( f . x = 1 & () . x > 0 ) ) holds
( - (ln * ()) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (ln * ())) | Z) . x = () / (1 - ()) ) )
proof end;

:: 23 f.x=exp_R(2*x)+1
theorem Th23: :: FDIFF_6:23
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((#Z 2) * exp_R) + f) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( ((#Z 2) * exp_R) + f is_differentiable_on Z & ( for x being Real st x in Z holds
((((#Z 2) * exp_R) + f) | Z) . x = 2 * (exp_R (2 * x)) ) )
proof end;

:: 24 f.x=(1/2)*ln(exp_R(2*x)+1)
theorem :: FDIFF_6:24
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) + f1 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) | Z) . x = () / (() + (exp_R (- x))) ) )
proof end;

:: 25 f.x=exp_R(2*x)-1
theorem Th25: :: FDIFF_6:25
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((#Z 2) * exp_R) - f) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( ((#Z 2) * exp_R) - f is_differentiable_on Z & ( for x being Real st x in Z holds
((((#Z 2) * exp_R) - f) | Z) . x = 2 * (exp_R (2 * x)) ) )
proof end;

:: 26 f.x=(1/2)*ln(exp_R(2*x)-1)
theorem :: FDIFF_6:26
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = ((#Z 2) * exp_R) - f1 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) holds
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) | Z) . x = () / (() - (exp_R (- x))) ) )
proof end;

:: 27 f.x=(exp_R.x-1)^2
theorem Th27: :: FDIFF_6:27
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * ()) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (#Z 2) * () is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * ()) | Z) . x = (2 * ()) * (() - 1) ) )
proof end;

:: 28 f.x=ln((exp_R.x-1)^2/exp_R.x)
theorem :: FDIFF_6:28
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) & ( for x being Real st x in Z holds
( f1 . x = 1 & (exp_R - f1) . x > 0 ) ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f | Z) . x = (() + 1) / (() - 1) ) )
proof end;

:: 29 f.x=(exp_R.x+1)^2
theorem Th29: :: FDIFF_6:29
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * ()) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (#Z 2) * () is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * ()) | Z) . x = (2 * ()) * (() + 1) ) )
proof end;

:: 30 f.x=ln((exp_R.x+1)^2/exp_R.x)
theorem :: FDIFF_6:30
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R + f1)) / exp_R) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f | Z) . x = (() - 1) / (() + 1) ) )
proof end;

:: 31 f.x=(1-exp_R.x)^2
theorem Th31: :: FDIFF_6:31
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * ()) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (#Z 2) * () is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * ()) | Z) . x = - ((2 * ()) * (1 - ())) ) )
proof end;

:: 32 f.x=ln(exp_R.x/(1-exp_R.x)^2)
theorem :: FDIFF_6:32
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 - exp_R))) & ( for x being Real st x in Z holds
( f1 . x = 1 & (f1 - exp_R) . x > 0 ) ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f | Z) . x = (1 + ()) / (1 - ()) ) )
proof end;

:: 33 f.x=ln(exp_R.x/(1+exp_R.x)^2)
theorem :: FDIFF_6:33
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R))) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f | Z) . x = (1 - ()) / (1 + ()) ) )
proof end;

:: 34 f.x=ln(exp_R(x)+exp_R(-x))
theorem :: FDIFF_6:34
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom (ln * f) & f = exp_R + (exp_R * f1) & ( for x being Real st x in Z holds
f1 . x = - x ) holds
( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) | Z) . x = (() - (exp_R (- x))) / (() + (exp_R (- x))) ) )
proof end;

:: 35 f.x=ln(exp_R(x)-exp_R(-x))
theorem :: FDIFF_6:35
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom (ln * f) & f = exp_R - (exp_R * f1) & ( for x being Real st x in Z holds
( f1 . x = - x & f . x > 0 ) ) holds
( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) | Z) . x = (() + (exp_R (- x))) / (() - (exp_R (- x))) ) )
proof end;

:: 36 f.x=(2/3)*((1+exp_R.x) ^ (3/2))
theorem :: FDIFF_6:36
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((2 / 3) (#) ((#R (3 / 2)) * ())) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (2 / 3) (#) ((#R (3 / 2)) * ()) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / 3) (#) ((#R (3 / 2)) * ())) | Z) . x = () * ((1 + ()) #R (1 / 2)) ) )
proof end;

:: 37 f.x=(2/(3*lna))*((1+a |^x) ^ (3/2))
theorem :: FDIFF_6:37
for a being Real
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) & ( for x being Real st x in Z holds
( f . x = 1 & f1 . x = x * (log (number_e,a)) ) ) & a > 0 & a <> 1 holds
( (2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1))) is_differentiable_on Z & ( for x being Real st x in Z holds
(((2 / (3 * (log (number_e,a)))) (#) ((#R (3 / 2)) * (f + (exp_R * f1)))) | Z) . x = (a #R x) * ((1 + (a #R x)) #R (1 / 2)) ) )
proof end;

:: 38 f.x=(-1/2)*cos(2*x)
theorem :: FDIFF_6:38
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((- (1 / 2)) (#) (cos * f)) & ( for x being Real st x in Z holds
f . x = 2 * x ) holds
( (- (1 / 2)) (#) (cos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (1 / 2)) (#) (cos * f)) | Z) . x = sin (2 * x) ) )
proof end;

theorem :: FDIFF_6:39
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) & ( for x being Real st x in Z holds
( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds
( 2 (#) ((#R (1 / 2)) * (f - cos)) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * (f - cos))) | Z) . x = (1 + (cos . x)) #R (1 / 2) ) )
proof end;

theorem :: FDIFF_6:40
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((- 2) (#) ((#R (1 / 2)) * (f + cos))) & ( for x being Real st x in Z holds
( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds
( (- 2) (#) ((#R (1 / 2)) * (f + cos)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- 2) (#) ((#R (1 / 2)) * (f + cos))) | Z) . x = (1 - (cos . x)) #R (1 / 2) ) )
proof end;

Lm6: for Z being open Subset of REAL
for f1 being PartFunc of REAL,REAL st Z c= dom (f1 + ()) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f1 + () is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ()) | Z) . x = 2 * (cos . x) ) )

proof end;

theorem :: FDIFF_6:41
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * f)) & f = f1 + () & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) holds
( (1 / 2) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * f)) | Z) . x = (cos . x) / (1 + (2 * (sin . x))) ) )
proof end;

Lm7: for Z being open Subset of REAL
for f1 being PartFunc of REAL,REAL st Z c= dom (f1 + ()) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f1 + () is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ()) | Z) . x = - (2 * (sin . x)) ) )

proof end;

theorem :: FDIFF_6:42
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((- (1 / 2)) (#) (ln * f)) & f = f1 + () & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) holds
( (- (1 / 2)) (#) (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (1 / 2)) (#) (ln * f)) | Z) . x = (sin . x) / (1 + (2 * (cos . x))) ) )
proof end;

:: 43 f.x=1/(4*a)*sin(2*a*x)
theorem Th43: :: FDIFF_6:43
for a being Real
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * a)) (#) (sin * f)) & ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 holds
( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) | Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) )
proof end;

:: 44 f.x=x/2-1/(4*a)*sin(2*a*x)
theorem :: FDIFF_6:44
for a being Real
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds
( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) | Z) . x = (sin (a * x)) ^2 ) )
proof end;

:: 45 f.x=x/2+1/(4*a)*sin(2*a*x)
theorem :: FDIFF_6:45
for a being Real
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds
( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) | Z) . x = (cos (a * x)) ^2 ) )
proof end;

:: 46 f.x=(1/n)*(cos.x ^ n)
theorem Th46: :: FDIFF_6:46
for n being Element of NAT
for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * cos)) & n > 0 holds
( (1 / n) (#) ((#Z n) * cos) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * cos)) | Z) . x = - (((cos . x) #Z (n - 1)) * (sin . x)) ) )
proof end;

:: 47 f.x=(1/3)*(cos.x ^ 3)-cos.x
theorem :: FDIFF_6:47
for Z being open Subset of REAL st Z c= dom (((1 / 3) (#) ((#Z 3) * cos)) - cos) holds
( ((1 / 3) (#) ((#Z 3) * cos)) - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / 3) (#) ((#Z 3) * cos)) - cos) | Z) . x = (sin . x) |^ 3 ) )
proof end;

:: 48 f.x=sin.x-(1/3)*(sin.x ^ 3)
theorem :: FDIFF_6:48
for Z being open Subset of REAL st Z c= dom (sin - ((1 / 3) (#) ((#Z 3) * sin))) holds
( sin - ((1 / 3) (#) ((#Z 3) * sin)) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin - ((1 / 3) (#) ((#Z 3) * sin))) | Z) . x = (cos . x) |^ 3 ) )
proof end;

:: 49 f.x=sin(ln x)
theorem :: FDIFF_6:49
for Z being open Subset of REAL st Z c= dom () holds
( sin * ln is_differentiable_on Z & ( for x being Real st x in Z holds
(() | Z) . x = (cos . (log (number_e,x))) / x ) )
proof end;

:: 50 f.x=-cos(ln x)
theorem :: FDIFF_6:50
for Z being open Subset of REAL st Z c= dom (- ()) holds
( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) | Z) . x = (sin . (log (number_e,x))) / x ) )
proof end;