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) )
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