Reciprocal Calculus

     Reciprocal Calculus

          Let D denote derivative and I denote integral; then define

          [1/D](f(x))    =    1 / D(1/f(x))    =    - f² / Df

          [1/I](f(x))     =  1  /  I(1/f(x))

          These are derivative and integral conjugated through reciprocal.

          Also define reciprocal addition and subtraction:

          x [+]  y   =   1 /  ( (1/x) + (1/y) )

          x [-]  y   =   1 /  ( (1/x) - (1/y) )

          Reciprocal addition is commutative, associative, has identity infinity, attractor zero, and multiplication distributes over it.

          These laws apply:

          [1/D]( f(x) [+] k )         =       [1/D]( f(x) )

          [1/D]( f(x) [+] g(x) )    =       [1/D]( f(x) )[+] [1/D]( g(x) )

          [1/D]( k * f(x) )  =       k * [1/D]( f(x) )

          [1/D]( x ^ n )                =       (-1/n) x ^ (n+1)

          [1/D]( e ^ (kx) )           =       (-1/k) e ^ (kx)

          [1/D]( 1 / ln(x) )           =       x

          [1/D]( csc(x) )              =       sec(x)

          [1/D]( sec(x) )              =       - csc(x)

          [1/D]( f(x) * g(x) )       =       f(x)*[1/D](g(x))  [+]   g(x)*[1/D](f(x))

          [1/I]( f(x) [+] g(x) )     =       [1/I](f(x))  [+]  [1/I](g(x))

          [1/I]( k * f(x) )             =       k * [1/I]( f(x) )

          [1/I]( x ^ n )                  =       (1-n)  x^(n-1)

          [1/I]( e ^ (kx) )             =       -k  e^(kx)

          [1/I]( x )                       =       1 / ln(x)

          [1/I]( csc(x) )                =       - sec(x)

          [1/I]( sec(x) )                =       csc(x)

          [1/I] ( f(x)*[1/D](g(x) )        

=       f(x)*g(x )  [-]  [1/I]( (g(x)*[1/D](f(x) )