Article FDIFF_8, MML version 4.99.1005
:: FDIFF_8:th 1
theorem
for b1 being Element of REAL
st b1 in dom tan
holds cos . b1 <> 0;
:: FDIFF_8:th 2
theorem
for b1 being Element of REAL
st b1 in dom cot
holds sin . b1 <> 0;
:: FDIFF_8:th 3
theorem
for b1 being natural set
for b2 being open Element of bool REAL
for b3, b4 being Function-like Relation of REAL,REAL
st b2 c= dom (b3 / b4)
for b5 being Element of REAL
st b5 in b2
holds ((b3 / b4) . b5) #Z b1 = ((b3 . b5) #Z b1) / ((b4 . b5) #Z b1);
:: FDIFF_8:th 4
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= dom (b4 / b5) &
(for b6 being Element of REAL
st b6 in b3
holds b4 . b6 = b6 + b1 & b5 . b6 = b6 - b2)
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_8:th 5
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom (ln * (b2 ^)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = 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_8:th 6
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 (tan * b4) &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds tan * b4 is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((tan * b4) `| b3) . b5 = b1 / ((cos . ((b1 * b5) + b2)) ^2));
:: FDIFF_8:th 7
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 (cot * b4) &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds cot * b4 is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((cot * b4) `| b3) . b5 = - (b1 / ((sin . ((b1 * b5) + b2)) ^2)));
:: FDIFF_8:th 8
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom (tan * (b2 ^)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3)
holds tan * (b2 ^) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((tan * (b2 ^)) `| b1) . b3 = - (1 / (b3 ^2 * ((cos . (1 / b3)) ^2))));
:: FDIFF_8:th 9
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom (cot * (b2 ^)) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3)
holds cot * (b2 ^) is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((cot * (b2 ^)) `| b1) . b3 = 1 / (b3 ^2 * ((sin . (1 / b3)) ^2)));
:: FDIFF_8:th 10
theorem
for b1, b2, b3 being Element of REAL
for b4 being open Element of bool REAL
for b5, b6 being Function-like Relation of REAL,REAL
st b4 c= dom (tan * (b5 + (b1 (#) b6))) &
b6 = #Z 2 &
(for b7 being Element of REAL
st b7 in b4
holds b5 . b7 = b2 + (b3 * b7))
holds tan * (b5 + (b1 (#) b6)) is_differentiable_on b4 &
(for b7 being Element of REAL
st b7 in b4
holds ((tan * (b5 + (b1 (#) b6))) `| b4) . b7 = (b3 + ((2 * b1) * b7)) / ((cos . ((b2 + (b3 * b7)) + (b1 * (b7 ^2)))) ^2));
:: FDIFF_8:th 11
theorem
for b1, b2, b3 being Element of REAL
for b4 being open Element of bool REAL
for b5, b6 being Function-like Relation of REAL,REAL
st b4 c= dom (cot * (b5 + (b1 (#) b6))) &
b6 = #Z 2 &
(for b7 being Element of REAL
st b7 in b4
holds b5 . b7 = b2 + (b3 * b7))
holds cot * (b5 + (b1 (#) b6)) is_differentiable_on b4 &
(for b7 being Element of REAL
st b7 in b4
holds ((cot * (b5 + (b1 (#) b6))) `| b4) . b7 = - ((b3 + ((2 * b1) * b7)) / ((sin . ((b2 + (b3 * b7)) + (b1 * (b7 ^2)))) ^2)));
:: FDIFF_8:th 12
theorem
for b1 being open Element of bool REAL
st b1 c= dom (tan * exp_R)
holds tan * exp_R is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((tan * exp_R) `| b1) . b2 = (exp_R . b2) / ((cos . (exp_R . b2)) ^2));
:: FDIFF_8:th 13
theorem
for b1 being open Element of bool REAL
st b1 c= dom (cot * exp_R)
holds cot * exp_R is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((cot * exp_R) `| b1) . b2 = - ((exp_R . b2) / ((sin . (exp_R . b2)) ^2)));
:: FDIFF_8:th 14
theorem
for b1 being open Element of bool REAL
st b1 c= dom (tan * ln)
holds tan * ln is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((tan * ln) `| b1) . b2 = 1 / (b2 * ((cos . (ln . b2)) ^2)));
:: FDIFF_8:th 15
theorem
for b1 being open Element of bool REAL
st b1 c= dom (cot * ln)
holds cot * ln is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((cot * ln) `| b1) . b2 = - (1 / (b2 * ((sin . (ln . b2)) ^2))));
:: FDIFF_8:th 16
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R * tan)
holds exp_R * tan is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R * tan) `| b1) . b2 = (exp_R . (tan . b2)) / ((cos . b2) ^2));
:: FDIFF_8:th 17
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R * cot)
holds exp_R * cot is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R * cot) `| b1) . b2 = - ((exp_R . (cot . b2)) / ((sin . b2) ^2)));
:: FDIFF_8:th 18
theorem
for b1 being open Element of bool REAL
st b1 c= dom (ln * tan)
holds ln * tan is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((ln * tan) `| b1) . b2 = 1 / ((cos . b2) * (sin . b2)));
:: FDIFF_8:th 19
theorem
for b1 being open Element of bool REAL
st b1 c= dom (ln * cot)
holds ln * cot is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((ln * cot) `| b1) . b2 = - (1 / ((sin . b2) * (cos . b2))));
:: FDIFF_8:th 20
theorem
for b1 being natural set
for b2 being open Element of bool REAL
st b2 c= dom ((#Z b1) * tan) & 1 <= b1
holds (#Z b1) * tan is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (((#Z b1) * tan) `| b2) . b3 = (b1 * ((sin . b3) #Z (b1 - 1))) / ((cos . b3) #Z (b1 + 1)));
:: FDIFF_8:th 21
theorem
for b1 being natural set
for b2 being open Element of bool REAL
st b2 c= dom ((#Z b1) * cot) & 1 <= b1
holds (#Z b1) * cot is_differentiable_on b2 &
(for b3 being Element of REAL
st b3 in b2
holds (((#Z b1) * cot) `| b2) . b3 = - ((b1 * ((cos . b3) #Z (b1 - 1))) / ((sin . b3) #Z (b1 + 1))));
:: FDIFF_8:th 22
theorem
for b1 being open Element of bool REAL
st b1 c= dom (tan + (cos ^)) &
(for b2 being Element of REAL
st b2 in b1
holds 1 + (sin . b2) <> 0 & 1 - (sin . b2) <> 0)
holds tan + (cos ^) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((tan + (cos ^)) `| b1) . b2 = 1 / (1 - (sin . b2)));
:: FDIFF_8:th 23
theorem
for b1 being open Element of bool REAL
st b1 c= dom (tan - (cos ^)) &
(for b2 being Element of REAL
st b2 in b1
holds 1 - (sin . b2) <> 0 & 1 + (sin . b2) <> 0)
holds tan - (cos ^) is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((tan - (cos ^)) `| b1) . b2 = 1 / (1 + (sin . b2)));
:: FDIFF_8:th 24
theorem
for b1 being open Element of bool REAL
st b1 c= dom (tan - id b1)
holds tan - id b1 is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((tan - id b1) `| b1) . b2 = (sin . b2) ^2 / ((cos . b2) ^2));
:: FDIFF_8:th 25
theorem
for b1 being open Element of bool REAL
st b1 c= dom ((- cot) - id b1)
holds (- cot) - id b1 is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds (((- cot) - id b1) `| b1) . b2 = (cos . b2) ^2 / ((sin . b2) ^2));
:: FDIFF_8: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 (((1 / b1) (#) (tan * b3)) - id b2) &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b1 * b4 & b1 <> 0)
holds ((1 / b1) (#) (tan * b3)) - id b2 is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds ((((1 / b1) (#) (tan * b3)) - id b2) `| b2) . b4 = (sin . (b1 * b4)) ^2 / ((cos . (b1 * b4)) ^2));
:: FDIFF_8:th 27
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 (((- (1 / b1)) (#) (cot * b3)) - id b2) &
(for b4 being Element of REAL
st b4 in b2
holds b3 . b4 = b1 * b4 & b1 <> 0)
holds ((- (1 / b1)) (#) (cot * b3)) - id b2 is_differentiable_on b2 &
(for b4 being Element of REAL
st b4 in b2
holds ((((- (1 / b1)) (#) (cot * b3)) - id b2) `| b2) . b4 = (cos . (b1 * b4)) ^2 / ((sin . (b1 * b4)) ^2));
:: FDIFF_8:th 28
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 (#) tan) &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds b4 (#) tan is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((b4 (#) tan) `| b3) . b5 = ((b1 * (sin . b5)) / (cos . b5)) + (((b1 * b5) + b2) / ((cos . b5) ^2)));
:: FDIFF_8:th 29
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 (#) cot) &
(for b5 being Element of REAL
st b5 in b3
holds b4 . b5 = (b1 * b5) + b2)
holds b4 (#) cot is_differentiable_on b3 &
(for b5 being Element of REAL
st b5 in b3
holds ((b4 (#) cot) `| b3) . b5 = ((b1 * (cos . b5)) / (sin . b5)) - (((b1 * b5) + b2) / ((sin . b5) ^2)));
:: FDIFF_8:th 30
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R (#) tan)
holds exp_R (#) tan is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R (#) tan) `| b1) . b2 = (((exp_R . b2) * (sin . b2)) / (cos . b2)) + ((exp_R . b2) / ((cos . b2) ^2)));
:: FDIFF_8:th 31
theorem
for b1 being open Element of bool REAL
st b1 c= dom (exp_R (#) cot)
holds exp_R (#) cot is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((exp_R (#) cot) `| b1) . b2 = (((exp_R . b2) * (cos . b2)) / (sin . b2)) - ((exp_R . b2) / ((sin . b2) ^2)));
:: FDIFF_8:th 32
theorem
for b1 being open Element of bool REAL
st b1 c= dom (ln (#) tan)
holds ln (#) tan is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((ln (#) tan) `| b1) . b2 = (((sin . b2) / (cos . b2)) / b2) + ((ln . b2) / ((cos . b2) ^2)));
:: FDIFF_8:th 33
theorem
for b1 being open Element of bool REAL
st b1 c= dom (ln (#) cot)
holds ln (#) cot is_differentiable_on b1 &
(for b2 being Element of REAL
st b2 in b1
holds ((ln (#) cot) `| b1) . b2 = (((cos . b2) / (sin . b2)) / b2) - ((ln . b2) / ((sin . b2) ^2)));
:: FDIFF_8:th 34
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom (b2 ^ (#) tan) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3)
holds b2 ^ (#) tan is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((b2 ^ (#) tan) `| b1) . b3 = (- (((sin . b3) / (cos . b3)) / (b3 ^2))) + ((1 / b3) / ((cos . b3) ^2)));
:: FDIFF_8:th 35
theorem
for b1 being open Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom (b2 ^ (#) cot) &
(for b3 being Element of REAL
st b3 in b1
holds b2 . b3 = b3)
holds b2 ^ (#) cot is_differentiable_on b1 &
(for b3 being Element of REAL
st b3 in b1
holds ((b2 ^ (#) cot) `| b1) . b3 = (- (((cos . b3) / (sin . b3)) / (b3 ^2))) - ((1 / b3) / ((sin . b3) ^2)));