Article FDIFF_10, MML version 4.99.1005

:: FDIFF_10:th 1
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (tan * cot)
   holds tan * cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((tan * cot) `| b1) . b2 = (1 / ((cos . (cot . b2)) ^2)) * - (1 / ((sin . b2) ^2)));

:: FDIFF_10:th 2
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (tan * tan)
   holds tan * tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((tan * tan) `| b1) . b2 = (1 / ((cos . (tan . b2)) ^2)) * (1 / ((cos . b2) ^2)));

:: FDIFF_10:th 3
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cot * cot)
   holds cot * cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cot * cot) `| b1) . b2 = (1 / ((sin . (cot . b2)) ^2)) * (1 / ((sin . b2) ^2)));

:: FDIFF_10:th 4
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cot * tan)
   holds cot * tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cot * tan) `| b1) . b2 = (- (1 / ((sin . (tan . b2)) ^2))) * (1 / ((cos . b2) ^2)));

:: FDIFF_10:th 5
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (tan - cot)
   holds tan - cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((tan - cot) `| b1) . b2 = (1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2)));

:: FDIFF_10:th 6
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (tan + cot)
   holds tan + cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((tan + cot) `| b1) . b2 = (1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2)));

:: FDIFF_10:th 7
theorem
for b1 being open Element of bool REAL holds
   sin * sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * sin) `| b1) . b2 = (cos . (sin . b2)) * (cos . b2));

:: FDIFF_10:th 8
theorem
for b1 being open Element of bool REAL holds
   sin * cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * cos) `| b1) . b2 = - ((cos . (cos . b2)) * (sin . b2)));

:: FDIFF_10:th 9
theorem
for b1 being open Element of bool REAL holds
   cos * sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * sin) `| b1) . b2 = - ((sin . (sin . b2)) * (cos . b2)));

:: FDIFF_10:th 10
theorem
for b1 being open Element of bool REAL holds
   cos * cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * cos) `| b1) . b2 = (sin . (cos . b2)) * (sin . b2));

:: FDIFF_10:th 11
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos (#) cot)
   holds cos (#) cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos (#) cot) `| b1) . b2 = (- (cos . b2)) - ((cos . b2) / ((sin . b2) ^2)));

:: FDIFF_10:th 12
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin (#) tan)
   holds sin (#) tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) tan) `| b1) . b2 = (sin . b2) + ((sin . b2) / ((cos . b2) ^2)));

:: FDIFF_10:th 13
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin (#) cot)
   holds sin (#) cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) cot) `| b1) . b2 = ((cos . b2) * (cot . b2)) - (1 / (sin . b2)));

:: FDIFF_10:th 14
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos (#) tan)
   holds cos (#) tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos (#) tan) `| b1) . b2 = (- ((sin . b2) ^2 / (cos . b2))) + (1 / (cos . b2)));

:: FDIFF_10:th 15
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) ^2 - ((sin . b2) ^2));

:: FDIFF_10:th 16
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln (#) sin)
   holds ln (#) sin is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln (#) sin) `| b1) . b2 = ((sin . b2) / b2) + ((ln . b2) * (cos . b2)));

:: FDIFF_10:th 17
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln (#) cos)
   holds ln (#) cos is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln (#) cos) `| b1) . b2 = ((cos . b2) / b2) - ((ln . b2) * (sin . b2)));

:: FDIFF_10:th 18
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln (#) exp_R)
   holds ln (#) exp_R is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln (#) exp_R) `| b1) . b2 = ((exp_R . b2) / b2) + ((ln . b2) * (exp_R . b2)));

:: FDIFF_10:th 19
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (ln * ln) &
         (for b2 being Element of REAL
               st b2 in b1
            holds 0 < b2)
   holds ln * ln is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((ln * ln) `| b1) . b2 = 1 / ((ln . b2) * b2));

:: FDIFF_10:th 20
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R * exp_R)
   holds exp_R * exp_R is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R * exp_R) `| b1) . b2 = (exp_R . (exp_R . b2)) * (exp_R . b2));

:: FDIFF_10:th 21
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin * tan)
   holds sin * tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * tan) `| b1) . b2 = (cos (tan . b2)) / ((cos . b2) ^2));

:: FDIFF_10:th 22
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin * cot)
   holds sin * cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * cot) `| b1) . b2 = - ((cos (cot . b2)) / ((sin . b2) ^2)));

:: FDIFF_10:th 23
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos * tan)
   holds cos * tan is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * tan) `| b1) . b2 = - ((sin (tan . b2)) / ((cos . b2) ^2)));

:: FDIFF_10:th 24
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos * cot)
   holds cos * cot is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * cot) `| b1) . b2 = (sin (cot . b2)) / ((sin . b2) ^2));

:: FDIFF_10:th 25
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin (#) (tan + cot))
   holds sin (#) (tan + cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) (tan + cot)) `| b1) . b2 = ((cos . b2) * ((tan . b2) + (cot . b2))) + ((sin . b2) * ((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 26
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos (#) (tan + cot))
   holds cos (#) (tan + cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos (#) (tan + cot)) `| b1) . b2 = (- ((sin . b2) * ((tan . b2) + (cot . b2)))) + ((cos . b2) * ((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 27
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin (#) (tan - cot))
   holds sin (#) (tan - cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) (tan - cot)) `| b1) . b2 = ((cos . b2) * ((tan . b2) - (cot . b2))) + ((sin . b2) * ((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 28
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos (#) (tan - cot))
   holds cos (#) (tan - cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos (#) (tan - cot)) `| b1) . b2 = (- ((sin . b2) * ((tan . b2) - (cot . b2)))) + ((cos . b2) * ((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 29
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R (#) (tan + cot))
   holds exp_R (#) (tan + cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R (#) (tan + cot)) `| b1) . b2 = ((exp_R . b2) * ((tan . b2) + (cot . b2))) + ((exp_R . b2) * ((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 30
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R (#) (tan - cot))
   holds exp_R (#) (tan - cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R (#) (tan - cot)) `| b1) . b2 = ((exp_R . b2) * ((tan . b2) - (cot . b2))) + ((exp_R . b2) * ((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 31
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin (#) (sin + cos))
   holds sin (#) (sin + cos) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) (sin + cos)) `| b1) . b2 = ((cos . b2) ^2 + ((2 * (sin . b2)) * (cos . b2))) - ((sin . b2) ^2));

:: FDIFF_10:th 32
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin (#) (sin - cos))
   holds sin (#) (sin - cos) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin (#) (sin - cos)) `| b1) . b2 = ((sin . b2) ^2 + ((2 * (sin . b2)) * (cos . b2))) - ((cos . b2) ^2));

:: FDIFF_10:th 33
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos (#) (sin - cos))
   holds cos (#) (sin - cos) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos (#) (sin - cos)) `| b1) . b2 = ((cos . b2) ^2 + ((2 * (sin . b2)) * (cos . b2))) - ((sin . b2) ^2));

:: FDIFF_10:th 34
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos (#) (sin + cos))
   holds cos (#) (sin + cos) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos (#) (sin + cos)) `| b1) . b2 = ((cos . b2) ^2 - ((2 * (sin . b2)) * (cos . b2))) - ((sin . b2) ^2));

:: FDIFF_10:th 35
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin * (tan + cot))
   holds sin * (tan + cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * (tan + cot)) `| b1) . b2 = (cos . ((tan . b2) + (cot . b2))) * ((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2))));

:: FDIFF_10:th 36
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin * (tan - cot))
   holds sin * (tan - cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * (tan - cot)) `| b1) . b2 = (cos . ((tan . b2) - (cot . b2))) * ((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2))));

:: FDIFF_10:th 37
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos * (tan - cot))
   holds cos * (tan - cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * (tan - cot)) `| b1) . b2 = - ((sin . ((tan . b2) - (cot . b2))) * ((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 38
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos * (tan + cot))
   holds cos * (tan + cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * (tan + cot)) `| b1) . b2 = - ((sin . ((tan . b2) + (cot . b2))) * ((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2)))));

:: FDIFF_10:th 39
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R * (tan + cot))
   holds exp_R * (tan + cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R * (tan + cot)) `| b1) . b2 = (exp_R . ((tan . b2) + (cot . b2))) * ((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2))));

:: FDIFF_10:th 40
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (exp_R * (tan - cot))
   holds exp_R * (tan - cot) is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((exp_R * (tan - cot)) `| b1) . b2 = (exp_R . ((tan . b2) - (cot . b2))) * ((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2))));

:: FDIFF_10:th 41
theorem
for b1 being open Element of bool REAL
      st b1 c= dom ((tan - cot) / exp_R)
   holds (tan - cot) / exp_R is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((tan - cot) / exp_R) `| b1) . b2 = ((((1 / ((cos . b2) ^2)) + (1 / ((sin . b2) ^2))) - (tan . b2)) + (cot . b2)) / (exp_R . b2));

:: FDIFF_10:th 42
theorem
for b1 being open Element of bool REAL
      st b1 c= dom ((tan + cot) / exp_R)
   holds (tan + cot) / exp_R is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds (((tan + cot) / exp_R) `| b1) . b2 = ((((1 / ((cos . b2) ^2)) - (1 / ((sin . b2) ^2))) - (tan . b2)) - (cot . b2)) / (exp_R . b2));

:: FDIFF_10:th 43
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin * sec)
   holds sin * sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * sec) `| b1) . b2 = ((cos . (sec . b2)) * (sin . b2)) / ((cos . b2) ^2));

:: FDIFF_10:th 44
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos * sec)
   holds cos * sec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * sec) `| b1) . b2 = - (((sin . (sec . b2)) * (sin . b2)) / ((cos . b2) ^2)));

:: FDIFF_10:th 45
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (sin * cosec)
   holds sin * cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((sin * cosec) `| b1) . b2 = - (((cos . (cosec . b2)) * (cos . b2)) / ((sin . b2) ^2)));

:: FDIFF_10:th 46
theorem
for b1 being open Element of bool REAL
      st b1 c= dom (cos * cosec)
   holds cos * cosec is_differentiable_on b1 &
    (for b2 being Element of REAL
          st b2 in b1
       holds ((cos * cosec) `| b1) . b2 = ((sin . (cosec . b2)) * (cos . b2)) / ((sin . b2) ^2));