Article FDIFF_5, MML version 4.99.1005
:: FDIFF_5:th 1
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
st b2 c= proj1 (b3 / b4) &
(for b5 being Element of REAL
st b5 in b2
holds b3 . b5 = b1 + b5 & b4 . b5 = b1 - b5 & b4 . b5 <> 0)
holds b3 / b4 is_differentiable_on b2 &
(for b5 being Element of REAL
st b5 in b2
holds ((b3 / b4) `| b2) . b5 = (2 * b1) / ((b1 - b5) ^2));
:: FDIFF_5:th 2
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
st b2 c= proj1 (b3 / b4) &
(for b5 being Element of REAL
st b5 in b2
holds b3 . b5 = b5 - b1 & b4 . b5 = b5 + b1 & b4 . b5 <> 0)
holds b3 / b4 is_differentiable_on b2 &
(for b5 being Element of REAL
st b5 in b2
holds ((b3 / b4) `| b2) . b5 = (2 * b1) / ((b5 + b1) ^2));
:: FDIFF_5:th 3
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4, b5 being Function-like Relation of REAL,REAL
st b3 c= proj1 (b4 / b5) &
(for b6 being Element of REAL
st b6 in b3
holds b4 . b6 = b6 - b1 & b5 . b6 = b6 - b2 & b5 . b6 <> 0)
holds b4 / b5 is_differentiable_on b3 &
(for b6 being Element of REAL
st b6 in b3
holds ((b4 / b5) `| b3) . b6 = (b1 - b2) / ((b6 - b2) ^2));
:: FDIFF_5:th 4
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 b2 &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds b2 ^ is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (b2 ^ `| b1) . b3 = - (1 / (b3 ^2)));
:: FDIFF_5:th 5
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 (sin * (b2 ^)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds sin * (b2 ^) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((sin * (b2 ^)) `| b1) . b3 = - ((1 / (b3 ^2)) * (cos . (1 / b3))));
:: FDIFF_5:th 6
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 (cos * (b2 ^)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds cos * (b2 ^) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((cos * (b2 ^)) `| b1) . b3 = (1 / (b3 ^2)) * (sin . (1 / b3)));
:: FDIFF_5:th 7
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 ((id b1) (#) (sin * (b2 ^))) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds (id b1) (#) (sin * (b2 ^)) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((id b1) (#) (sin * (b2 ^))) `| b1) . b3 = (sin . (1 / b3)) - ((1 / b3) * (cos . (1 / b3))));
:: FDIFF_5:th 8
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 ((id b1) (#) (cos * (b2 ^))) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds (id b1) (#) (cos * (b2 ^)) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((id b1) (#) (cos * (b2 ^))) `| b1) . b3 = (cos . (1 / b3)) + ((1 / b3) * (sin . (1 / b3))));
:: FDIFF_5:th 9
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 ((sin * (b2 ^)) (#) (cos * (b2 ^))) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds (sin * (b2 ^)) (#) (cos * (b2 ^)) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((sin * (b2 ^)) (#) (cos * (b2 ^))) `| b1) . b3 = (1 / (b3 ^2)) * ((sin . (1 / b3)) ^2 - ((cos . (1 / b3)) ^2)));
:: FDIFF_5:th 10
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= proj1 ((sin * b3) (#) ((#Z b1) * sin)) &
1 <= b1 &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b1 * b4)
holds (sin * b3) (#) ((#Z b1) * sin) is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds (((sin * b3) (#) ((#Z b1) * sin)) `| b2) . b4 = (b1 * ((sin . b4) #Z (b1 - 1))) * (sin . ((b1 + 1) * b4)));
:: FDIFF_5:th 11
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= proj1 ((cos * b3) (#) ((#Z b1) * sin)) &
1 <= b1 &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b1 * b4)
holds (cos * b3) (#) ((#Z b1) * sin) is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds (((cos * b3) (#) ((#Z b1) * sin)) `| b2) . b4 = (b1 * ((sin . b4) #Z (b1 - 1))) * (cos . ((b1 + 1) * b4)));
:: FDIFF_5:th 12
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= proj1 ((cos * b3) (#) ((#Z b1) * cos)) &
1 <= b1 &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b1 * b4)
holds (cos * b3) (#) ((#Z b1) * cos) is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds (((cos * b3) (#) ((#Z b1) * cos)) `| b2) . b4 = - ((b1 * ((cos . b4) #Z (b1 - 1))) * (sin . ((b1 + 1) * b4))));
:: FDIFF_5:th 13
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= proj1 ((sin * b3) (#) ((#Z b1) * cos)) &
1 <= b1 &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b1 * b4)
holds (sin * b3) (#) ((#Z b1) * cos) is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds (((sin * b3) (#) ((#Z b1) * cos)) `| b2) . b4 = (b1 * ((cos . b4) #Z (b1 - 1))) * (cos . ((b1 + 1) * b4)));
:: FDIFF_5:th 14
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 ^ (#) sin) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds b2 ^ (#) sin is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((b2 ^ (#) sin) `| b1) . b3 = ((1 / b3) * (cos . b3)) - ((1 / (b3 ^2)) * (sin . b3)));
:: FDIFF_5:th 15
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 ^ (#) cos) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & b2 . b3 <> 0)
holds b2 ^ (#) cos is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((b2 ^ (#) cos) `| b1) . b3 = (- ((1 / b3) * (sin . b3))) - ((1 / (b3 ^2)) * (cos . b3)));
:: FDIFF_5:th 16
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 (sin + ((#R (1 / 2)) * b2)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & 0 < b2 . b3)
holds sin + ((#R (1 / 2)) * b2) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((sin + ((#R (1 / 2)) * b2)) `| b1) . b3 = (cos . b3) + ((1 / 2) * (b3 #R - (1 / 2))));
:: FDIFF_5:th 17
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 (#) (sin * (b3 ^))) &
b2 = #Z 2 &
(for b4 being Element of REAL
st b4 in b1
holds b3 . b4 = b4 & b3 . b4 <> 0)
holds b2 (#) (sin * (b3 ^)) is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds ((b2 (#) (sin * (b3 ^))) `| b1) . b4 = ((2 * b4) * (sin . (1 / b4))) - (cos . (1 / b4)));
:: FDIFF_5:th 18
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 (#) (cos * (b3 ^))) &
b2 = #Z 2 &
(for b4 being Element of REAL
st b4 in b1
holds b3 . b4 = b4 & b3 . b4 <> 0)
holds b2 (#) (cos * (b3 ^)) is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds ((b2 (#) (cos * (b3 ^))) `| b1) . b4 = ((2 * b4) * (cos . (1 / b4))) + (sin . (1 / b4)));
:: FDIFF_5:th 19
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 (ln * b2) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 & 0 < b2 . b3)
holds ln * b2 is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((ln * b2) `| b1) . b3 = 1 / b3);
:: FDIFF_5:th 20
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= proj1 ((id b1) (#) b2) &
b2 = ln * b3 &
(for b4 being Element of REAL
st b4 in b1
holds b3 . b4 = b4 & 0 < b3 . b4)
holds (id b1) (#) b2 is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds (((id b1) (#) b2) `| b1) . b4 = 1 + (ln . b4));
:: FDIFF_5:th 21
theorem
for b1 being open Element of bool REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 (#) b3) &
b2 = #Z 2 &
b3 = ln * b4 &
(for b5 being Element of REAL
st b5 in b1
holds b4 . b5 = b5 & 0 < b4 . b5)
holds b2 (#) b3 is_differentiable_on b1 &
(for b5 being Element of REAL
st b5 in b1
holds ((b2 (#) b3) `| b1) . b5 = b5 + ((2 * b5) * (ln . b5)));
:: FDIFF_5:th 22
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
st b2 c= proj1 ((b3 + b4) / (b3 - b4)) &
(for b5 being Element of REAL
st b5 in b2
holds b3 . b5 = b1) &
b4 = #Z 2 &
(for b5 being Element of REAL
st b5 in b2
holds 0 < (b3 - b4) . b5)
holds (b3 + b4) / (b3 - b4) is_differentiable_on b2 &
(for b5 being Element of REAL
st b5 in b2
holds (((b3 + b4) / (b3 - b4)) `| b2) . b5 = ((4 * b1) * b5) / ((b1 - (b5 |^ 2)) |^ 2));
:: FDIFF_5:th 23
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
st b2 c= proj1 (ln * ((b3 + b4) / (b3 - b4))) &
(for b5 being Element of REAL
st b5 in b2
holds b3 . b5 = b1) &
b4 = #Z 2 &
(for b5 being Element of REAL
st b5 in b2
holds 0 < (b3 - b4) . b5) &
(for b5 being Element of REAL
st b5 in b2
holds 0 < (b3 + b4) . b5)
holds ln * ((b3 + b4) / (b3 - b4)) is_differentiable_on b2 &
(for b5 being Element of REAL
st b5 in b2
holds ((ln * ((b3 + b4) / (b3 - b4))) `| b2) . b5 = ((4 * b1) * b5) / ((b1 |^ 2) - (b5 |^ 4)));
:: FDIFF_5:th 24
theorem
for b1 being open Element of bool REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 ^ (#) b3) &
(for b5 being Element of REAL
st b5 in b1
holds b2 . b5 = b5) &
b3 = ln * b4 &
(for b5 being Element of REAL
st b5 in b1
holds b4 . b5 = b5 & 0 < b4 . b5)
holds b2 ^ (#) b3 is_differentiable_on b1 &
(for b5 being Element of REAL
st b5 in b1
holds ((b2 ^ (#) b3) `| b1) . b5 = (1 / (b5 ^2)) * (1 - (ln . b5)));
:: FDIFF_5:th 25
theorem
for b1 being open Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= proj1 (b2 ^) &
b2 = ln * b3 &
(for b4 being Element of REAL
st b4 in b1
holds b3 . b4 = b4 & 0 < b3 . b4) &
(for b4 being Element of REAL
st b4 in b1
holds b2 . b4 <> 0)
holds b2 ^ is_differentiable_on b1 &
(for b4 being Element of REAL
st b4 in b1
holds (b2 ^ `| b1) . b4 = - (1 / (b4 * ((ln . b4) ^2))));