Article FDIFF_7, MML version 4.99.1005
:: FDIFF_7:th 1
theorem
for b1 being Element of REAL holds
b1 #Z 2 = b1 ^2;
:: FDIFF_7:th 2
theorem
for b1 being Element of REAL
st 0 < b1
holds b1 #R (1 / 2) = sqrt b1;
:: FDIFF_7:th 3
theorem
for b1 being Element of REAL
st 0 < b1
holds b1 #R - (1 / 2) = 1 / sqrt b1;
:: FDIFF_7:th 4
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
st b2 c= ].- 1,1.[ & b2 c= dom (b1 (#) arcsin)
holds b1 (#) arcsin is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds ((b1 (#) arcsin) `| b2) . b3 = b1 / sqrt (1 - (b3 ^2)));
:: FDIFF_7:th 5
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
st b2 c= ].- 1,1.[ & b2 c= dom (b1 (#) arccos)
holds b1 (#) arccos is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds ((b1 (#) arccos) `| b2) . b3 = - (b1 / sqrt (1 - (b3 ^2))));
:: FDIFF_7:th 6
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_differentiable_in b1 & - 1 < b2 . b1 & b2 . b1 < 1
holds arcsin * b2 is_differentiable_in b1 &
diff(arcsin * b2,b1) = (diff(b2,b1)) / sqrt (1 - ((b2 . b1) ^2));
:: FDIFF_7:th 7
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_differentiable_in b1 & - 1 < b2 . b1 & b2 . b1 < 1
holds arccos * b2 is_differentiable_in b1 &
diff(arccos * b2,b1) = - ((diff(b2,b1)) / sqrt (1 - ((b2 . b1) ^2)));
:: FDIFF_7:th 8
theorem
for b1 being open Element of bool REAL
st b1 c= dom (ln * arcsin) &
b1 c= ].- 1,1.[ &
(for b2 being Element of REAL
st b2 in b1
holds 0 < arcsin . b2)
holds ln * arcsin is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((ln * arcsin) `| b1) . b2 = 1 / ((sqrt (1 - (b2 ^2))) * (arcsin . b2)));
:: FDIFF_7:th 9
theorem
for b1 being open Element of bool REAL
st b1 c= dom (ln * arccos) &
b1 c= ].- 1,1.[ &
(for b2 being Element of REAL
st b2 in b1
holds 0 < arccos . b2)
holds ln * arccos is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((ln * arccos) `| b1) . b2 = - (1 / ((sqrt (1 - (b2 ^2))) * (arccos . b2))));
:: FDIFF_7:th 10
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
st b2 c= dom ((#Z b1) * arcsin) & b2 c= ].- 1,1.[
holds (#Z b1) * arcsin is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (((#Z b1) * arcsin) `| b2) . b3 = (b1 * ((arcsin . b3) #Z (b1 - 1))) / sqrt (1 - (b3 ^2)));
:: FDIFF_7:th 11
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
st b2 c= dom ((#Z b1) * arccos) & b2 c= ].- 1,1.[
holds (#Z b1) * arccos is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (((#Z b1) * arccos) `| b2) . b3 = - ((b1 * ((arccos . b3) #Z (b1 - 1))) / sqrt (1 - (b3 ^2))));
:: FDIFF_7:th 12
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) &
b1 c= ].- 1,1.[
holds (1 / 2) (#) ((#Z 2) * arcsin) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((1 / 2) (#) ((#Z 2) * arcsin)) `| b1) . b2 = (arcsin . b2) / sqrt (1 - (b2 ^2)));
:: FDIFF_7:th 13
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((1 / 2) (#) ((#Z 2) * arccos)) &
b1 c= ].- 1,1.[
holds (1 / 2) (#) ((#Z 2) * arccos) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((1 / 2) (#) ((#Z 2) * arccos)) `| b1) . b2 = - ((arccos . b2) / sqrt (1 - (b2 ^2))));
:: FDIFF_7:th 14
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
st b3 c= dom (arcsin * b4) &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2 & - 1 < b4 . b5 & b4 . b5 < 1)
holds arcsin * b4 is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((arcsin * b4) `| b3) . b5 = b1 / sqrt (1 - (((b1 * b5) + b2) ^2)));
:: FDIFF_7:th 15
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
st b3 c= dom (arccos * b4) &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2 & - 1 < b4 . b5 & b4 . b5 < 1)
holds arccos * b4 is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((arccos * b4) `| b3) . b5 = - (b1 / sqrt (1 - (((b1 * b5) + b2) ^2))));
:: FDIFF_7:th 16
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((id b1) (#) arcsin) & b1 c= ].- 1,1.[
holds (id b1) (#) arcsin is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((id b1) (#) arcsin) `| b1) . b2 = (arcsin . b2) + (b2 / sqrt (1 - (b2 ^2))));
:: FDIFF_7:th 17
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((id b1) (#) arccos) & b1 c= ].- 1,1.[
holds (id b1) (#) arccos is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((id b1) (#) arccos) `| b1) . b2 = (arccos . b2) - (b2 / sqrt (1 - (b2 ^2))));
:: FDIFF_7:th 18
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
st b3 c= dom (b4 (#) arcsin) &
b3 c= ].- 1,1.[ &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds b4 (#) arcsin is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((b4 (#) arcsin) `| b3) . b5 = (b1 * (arcsin . b5)) + (((b1 * b5) + b2) / sqrt (1 - (b5 ^2))));
:: FDIFF_7:th 19
theorem
for b1, b2 being Element of REAL
for b3 being open Element of bool REAL
for b4 being Function-like Relation of REAL,REAL
st b3 c= dom (b4 (#) arccos) &
b3 c= ].- 1,1.[ &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds b4 (#) arccos is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((b4 (#) arccos) `| b3) . b5 = (b1 * (arccos . b5)) - (((b1 * b5) + b2) / sqrt (1 - (b5 ^2))));
:: FDIFF_7:th 20
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom ((1 / 2) (#) (arcsin * b2)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = 2 * b3 & - 1 < b2 . b3 & b2 . b3 < 1)
holds (1 / 2) (#) (arcsin * b2) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((1 / 2) (#) (arcsin * b2)) `| b1) . b3 = 1 / sqrt (1 - ((2 * b3) ^2)));
:: FDIFF_7:th 21
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom ((1 / 2) (#) (arccos * b2)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = 2 * b3 & - 1 < b2 . b3 & b2 . b3 < 1)
holds (1 / 2) (#) (arccos * b2) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((1 / 2) (#) (arccos * b2)) `| b1) . b3 = - (1 / sqrt (1 - ((2 * b3) ^2))));
:: FDIFF_7:th 22
theorem
for b1 being open Element of bool REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b1 c= dom ((#R (1 / 2)) * b2) &
b2 = b3 - b4 &
b4 = #Z 2 &
(for b5 being Element of REAL
st b5 in b1
holds b3 . b5 = 1 & 0 < b2 . b5)
holds (#R (1 / 2)) * b2 is_differentiable_on b1 &
(for b5 being Element of REAL
st b5 in b1
holds (((#R (1 / 2)) * b2) `| b1) . b5 = - (b5 * ((1 - (b5 #Z 2)) #R - (1 / 2))));
:: FDIFF_7:th 23
theorem
for b1 being open Element of bool REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b1 c= dom (((id b1) (#) arcsin) + ((#R (1 / 2)) * b2)) &
b1 c= ].- 1,1.[ &
b2 = b3 - b4 &
b4 = #Z 2 &
(for b5 being Element of REAL
st b5 in b1
holds b3 . b5 = 1 & 0 < b2 . b5 & b5 <> 0)
holds ((id b1) (#) arcsin) + ((#R (1 / 2)) * b2) is_differentiable_on b1 &
(for b5 being Element of REAL
st b5 in b1
holds ((((id b1) (#) arcsin) + ((#R (1 / 2)) * b2)) `| b1) . b5 = arcsin . b5);
:: FDIFF_7: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= dom (((id b1) (#) arccos) - ((#R (1 / 2)) * b2)) &
b1 c= ].- 1,1.[ &
b2 = b3 - b4 &
b4 = #Z 2 &
(for b5 being Element of REAL
st b5 in b1
holds b3 . b5 = 1 & 0 < b2 . b5 & b5 <> 0)
holds ((id b1) (#) arccos) - ((#R (1 / 2)) * b2) is_differentiable_on b1 &
(for b5 being Element of REAL
st b5 in b1
holds ((((id b1) (#) arccos) - ((#R (1 / 2)) * b2)) `| b1) . b5 = arccos . b5);
:: FDIFF_7:th 25
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= dom ((id b2) (#) (arcsin * b3)) &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b4 / b1 & - 1 < b3 . b4 & b3 . b4 < 1)
holds (id b2) (#) (arcsin * b3) is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds (((id b2) (#) (arcsin * b3)) `| b2) . b4 = (arcsin . (b4 / b1)) + (b4 / (b1 * sqrt (1 - ((b4 / b1) ^2)))));
:: FDIFF_7:th 26
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3 being Function-like Relation of REAL,REAL
st b2 c= dom ((id b2) (#) (arccos * b3)) &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b4 / b1 & - 1 < b3 . b4 & b3 . b4 < 1)
holds (id b2) (#) (arccos * b3) is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds (((id b2) (#) (arccos * b3)) `| b2) . b4 = (arccos . (b4 / b1)) - (b4 / (b1 * sqrt (1 - ((b4 / b1) ^2)))));
:: FDIFF_7:th 27
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4, b5 being Function-like Relation of REAL,REAL
st b2 c= dom ((#R (1 / 2)) * b3) &
b3 = b4 - b5 &
b5 = #Z 2 &
(for b6 being Element of REAL
st b6 in b2
holds b4 . b6 = b1 ^2 & 0 < b3 . b6)
holds (#R (1 / 2)) * b3 is_differentiable_on b2 &
(for b6 being Element of REAL
st b6 in b2
holds (((#R (1 / 2)) * b3) `| b2) . b6 = - (b6 * ((b1 ^2 - (b6 #Z 2)) #R - (1 / 2))));
:: FDIFF_7:th 28
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4, b5, b6 being Function-like Relation of REAL,REAL
st b2 c= dom (((id b2) (#) (arcsin * b3)) + ((#R (1 / 2)) * b4)) &
b2 c= ].- 1,1.[ &
b4 = b5 - b6 &
b6 = #Z 2 &
(for b7 being Element of REAL
st b7 in b2
holds b5 . b7 = b1 ^2 & 0 < b4 . b7 & b3 . b7 = b7 / b1 & - 1 < b3 . b7 & b3 . b7 < 1 & b7 <> 0 & 0 < b1)
holds ((id b2) (#) (arcsin * b3)) + ((#R (1 / 2)) * b4) is_differentiable_on b2 &
(for b7 being Element of REAL
st b7 in b2
holds ((((id b2) (#) (arcsin * b3)) + ((#R (1 / 2)) * b4)) `| b2) . b7 = arcsin . (b7 / b1));
:: FDIFF_7:th 29
theorem
for b1 being Element of REAL
for b2 being open Element of bool REAL
for b3, b4, b5, b6 being Function-like Relation of REAL,REAL
st b2 c= dom (((id b2) (#) (arccos * b3)) - ((#R (1 / 2)) * b4)) &
b2 c= ].- 1,1.[ &
b4 = b5 - b6 &
b6 = #Z 2 &
(for b7 being Element of REAL
st b7 in b2
holds b5 . b7 = b1 ^2 & 0 < b4 . b7 & b3 . b7 = b7 / b1 & - 1 < b3 . b7 & b3 . b7 < 1 & b7 <> 0 & 0 < b1)
holds ((id b2) (#) (arccos * b3)) - ((#R (1 / 2)) * b4) is_differentiable_on b2 &
(for b7 being Element of REAL
st b7 in b2
holds ((((id b2) (#) (arccos * b3)) - ((#R (1 / 2)) * b4)) `| b2) . b7 = arccos . (b7 / b1));
:: FDIFF_7:th 30
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
st b2 c= dom ((- (1 / b1)) (#) ((#Z b1) * (sin ^))) &
0 < b1 &
(for b3 being Element of REAL
st b3 in b2
holds sin . b3 <> 0)
holds (- (1 / b1)) (#) ((#Z b1) * (sin ^)) is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (((- (1 / b1)) (#) ((#Z b1) * (sin ^))) `| b2) . b3 = (cos . b3) / ((sin . b3) #Z (b1 + 1)));
:: FDIFF_7:th 31
theorem
for b1 being Element of NAT
for b2 being open Element of bool REAL
st b2 c= dom ((1 / b1) (#) ((#Z b1) * (cos ^))) &
0 < b1 &
(for b3 being Element of REAL
st b3 in b2
holds cos . b3 <> 0)
holds (1 / b1) (#) ((#Z b1) * (cos ^)) is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (((1 / b1) (#) ((#Z b1) * (cos ^))) `| b2) . b3 = (sin . b3) / ((cos . b3) #Z (b1 + 1)));
:: FDIFF_7:th 32
theorem
for b1 being open Element of bool REAL
st b1 c= dom (sin * ln) &
(for b2 being Element of REAL
st b2 in b1
holds 0 < b2)
holds sin * ln is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((sin * ln) `| b1) . b2 = (cos . (ln . b2)) / b2);
:: FDIFF_7:th 33
theorem
for b1 being open Element of bool REAL
st b1 c= dom (cos * ln) &
(for b2 being Element of REAL
st b2 in b1
holds 0 < b2)
holds cos * ln is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((cos * ln) `| b1) . b2 = - ((sin . (ln . b2)) / b2));
:: FDIFF_7:th 34
theorem
for b1 being open Element of bool REAL
st b1 c= dom (sin * exp_R)
holds sin * exp_R is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((sin * exp_R) `| b1) . b2 = (exp_R . b2) * (cos . (exp_R . b2)));
:: FDIFF_7:th 35
theorem
for b1 being open Element of bool REAL
st b1 c= dom (cos * exp_R)
holds cos * exp_R is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((cos * exp_R) `| b1) . b2 = - ((exp_R . b2) * (sin . (exp_R . b2))));
:: FDIFF_7:th 36
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R * cos)
holds exp_R * cos is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R * cos) `| b1) . b2 = - ((exp_R . (cos . b2)) * (sin . b2)));
:: FDIFF_7:th 37
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R * sin)
holds exp_R * sin is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R * sin) `| b1) . b2 = (exp_R . (sin . b2)) * (cos . b2));
:: FDIFF_7:th 38
theorem
for b1 being open Element of bool REAL
st b1 c= dom (sin + cos)
holds sin + cos is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((sin + cos) `| b1) . b2 = (cos . b2) - (sin . b2));
:: FDIFF_7:th 39
theorem
for b1 being open Element of bool REAL
st b1 c= dom (sin - cos)
holds sin - cos is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((sin - cos) `| b1) . b2 = (cos . b2) + (sin . b2));
:: FDIFF_7:th 40
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R (#) (sin - cos))
holds exp_R (#) (sin - cos) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R (#) (sin - cos)) `| b1) . b2 = (2 * (exp_R . b2)) * (sin . b2));
:: FDIFF_7:th 41
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R (#) (sin + cos))
holds exp_R (#) (sin + cos) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R (#) (sin + cos)) `| b1) . b2 = (2 * (exp_R . b2)) * (cos . b2));
:: FDIFF_7:th 42
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((sin + cos) / exp_R)
holds (sin + cos) / exp_R is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((sin + cos) / exp_R) `| b1) . b2 = - ((2 * (sin . b2)) / (exp_R . b2)));
:: FDIFF_7:th 43
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((sin - cos) / exp_R)
holds (sin - cos) / exp_R is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((sin - cos) / exp_R) `| b1) . b2 = (2 * (cos . b2)) / (exp_R . b2));
:: FDIFF_7:th 44
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R (#) sin)
holds exp_R (#) sin is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R (#) sin) `| b1) . b2 = (exp_R . b2) * ((sin . b2) + (cos . b2)));
:: FDIFF_7:th 45
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R (#) cos)
holds exp_R (#) cos is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R (#) cos) `| b1) . b2 = (exp_R . b2) * ((cos . b2) - (sin . b2)));
:: FDIFF_7:th 46
theorem
for b1 being Element of REAL
st cos . b1 <> 0
holds sin / cos is_differentiable_in b1 &
diff(sin / cos,b1) = 1 / ((cos . b1) ^2);
:: FDIFF_7:th 47
theorem
for b1 being Element of REAL
st sin . b1 <> 0
holds cos / sin is_differentiable_in b1 &
diff(cos / sin,b1) = - (1 / ((sin . b1) ^2));
:: FDIFF_7:th 48
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((#Z 2) * (sin / cos)) &
(for b2 being Element of REAL
st b2 in b1
holds cos . b2 <> 0)
holds (#Z 2) * (sin / cos) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((#Z 2) * (sin / cos)) `| b1) . b2 = (2 * (sin . b2)) / ((cos . b2) #Z 3));
:: FDIFF_7:th 49
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((#Z 2) * (cos / sin)) &
(for b2 being Element of REAL
st b2 in b1
holds sin . b2 <> 0)
holds (#Z 2) * (cos / sin) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((#Z 2) * (cos / sin)) `| b1) . b2 = - ((2 * (cos . b2)) / ((sin . b2) #Z 3)));
:: FDIFF_7:th 50
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom ((sin / cos) * b2) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 / 2 & cos . (b2 . b3) <> 0)
holds (sin / cos) * b2 is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((sin / cos) * b2) `| b1) . b3 = 1 / (1 + (cos . b3)));
:: FDIFF_7:th 51
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom ((cos / sin) * b2) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3 / 2 & sin . (b2 . b3) <> 0)
holds (cos / sin) * b2 is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds (((cos / sin) * b2) `| b1) . b3 = - (1 / (1 - (cos . b3))));