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)));