```Differentiation
---------------
Differentiation  is achieved using just another expression transformation with a
set of reduction rules, located towards the end of →#.math←. "∆x" can be read as
"dx". For example, the product rule D(u*v) = uD(v) + vD(u) is expressed:

(p*q)∆a  =  p∆a * q  +  p * q∆a

Pattern matching variables a, b, c match expression variables x and t, as in the
expression: (3*x∧2 + 2*x + 1)∆x, or (t∧2)∆t.

The  name '∆' has no special significance to the code that implements reduction;
it is just another function defined as having high precedence and binding left.

← ∆()                       ⍝ derivative wrt ⍵

Left binding means that we can write a second differential without parentheses:

until''
(1/t)∆t∆t
2*t∧-3

Symbol  'd'  could  equally well have been used for the operator, although there
would be a danger of confusion caused by errors such as:

a+b+c+d, a=1, b=2, c=3, d=4
\ missing operand

Donald  Knuth says: "Programs for algebraic differentiation were among the first
symbol-manipulation routines ever written for computers; they were used as early
as 1952".

The  reduction process is vital to the rules that define differentiation; other-
wise,  unmanageably  large  expressions result. Even so, there is much scope for
improvement as the following third derivative shows:

eval'(x∧x∧x)∆x∆x∆x'         ⍝ 3rd derivative in linear form
((ln(x)+1)*x∧x+1)*x∧(x+x∧x-2)+(x+x∧x-1)*((x+x∧x-2)*x∧(x+x∧x-3)+((ln(x)+1)*x∧x+1)
*ln(x)*x∧(x+x∧x-2))+((x∧(x-1)+x∧x*(ln(x)+1)∧2)*ln(x)+((ln(x)+1)*x∧x+1)/x)*
x∧(x+x∧x-1)+((ln(x)+1)*x∧x+1)*ln(x)*((x+x∧x-1)*x∧(x+x∧x-2)+((ln(x)+1)*x∧x+
1)*ln(x)*x∧(x+x∧x-1))+(((x-1)*x∧(x-2)+ln(x)*x∧(x-1)+(ln(x)+1)*x∧x*(ln(x)+1
)∧2+2*(x∧x*((ln(x)+1)/x)))*ln(x)+(x∧(x-1)+x∧x*(ln(x)+1)∧2)/x+((x∧(x-1)+x∧x
*(ln(x)+1)∧2)/x-(ln(x)+1)*x∧x/x∧2))*x∧x∧x+2*(((x∧(x-1)+x∧x*(ln(x)+1)∧2)*ln
(x)+(ln(x)+1)*x∧x/x)*(x∧(x+x∧x-1)+(ln(x)+1)*x∧x*ln(x)*x∧x∧x))+(ln(x)+1)*x∧
x*ln(x)*((x+x∧x-1)*x∧(x+x∧x-2)+((ln(x)+1)*x∧x+1)*ln(x)*x∧(x+x∧x-1)+((x∧(x-
1)+x∧x*(ln(x)+1)∧2)*ln(x)+(ln(x)+1)*x∧x/x)*x∧x∧x+(ln(x)+1)*x∧x*ln(x)*(x∧(x
+x∧x-1)+(ln(x)+1)*x∧x*ln(x)*x∧x∧x))

't' eval '(x∧x∧x)∆x∆x∆x'  ⍝ 3rd derivative in tree form
+
┌────────────────────────────────────────────────────────────┴───────────┐
+                                                                        *
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────┴─┐                                                                   ┌──┴──────────────────────────────────────────────────────────────────────────────┐
+                                                                                                             *                                                                   *                                                                                 +
┌───────────────────────────────────────────────────────┴─────────────────────────────────────────────────────────────────────────────────────────────────────┐      ┌┴──────────────────────────────┐                                ┌───┴─┐                                          ┌────────────────────────────────────┴───────────┐
+                                                                                                                                                             *      2                               *                                *    ln                                          +                                                *
┌──────────────────────────────────────────┴─────────────┐                                                                                                               ┌───────────────────────────────┴─┐                         ┌──────────┴─────────┐                    ┌─┴─┐  └┐                  ┌───────────────────────┴──────────────────────────────┐              ┌──┴─────────┐
+                                                        *                                                                                                               +                                 ∧                         +                    +                    +   ∧   x                  +                                                      *              *            +
┌─────────────────────────────────────────────────┴────────────────────────────────┐                    ┌──┴─────────────────┐                                                                          ┌──────────────────┴──────────────────┐             ┌┴─┐                    ┌──┴────────┐   ┌───────┴───────────┐      ┌─┴┐ ┌┴┐           ┌─────────┴─────────────┐                             ┌──────────┴─┐        ┌───┴─┐  ┌───────┴───────────┐
+                                                                                  *                    *                    +                                                                          +                                     -             x  ∧                    *           /   ∧                   *      ln 1 x x           *                       *                             +            ∧        *    ln  ∧                   *
┌─────────┴───────┐                                                             ┌────────────┴─┐                ┌─┴─┐        ┌─────────┴─────────────┐                                                         ┌──┴────────────────┐                  ┌─┴────────┐      ┌┴┐        ┌──────────┴─┐     ┌───┴┐ ┌┴─────┐          ┌──┴─┐    └┐               ┌─┴─┐                  ┌──┴─┐                        ┌──┴────────┐  ┌┴─┐    ┌─┴─┐  └┐ ┌┴─────┐          ┌──┴─┐
*                 *                                                             +              ∧                +  ln        *                       *                                                         *                   /                  /          /      x x        +           ln     *    x x      -          *    ∧     x               -   ∧                  *    ∧                        *           /  x  ∧    +   ∧   x x      -          *    ∧
┌─┴─┐             ┌─┴─────────────────┐                                        ┌──┴──────────┐  ┌┴─────┐      ┌───┴┐ └┐      ┌─┴─┐                  ┌──┴─┐                                          ┌────────────┴─┐      ┌──────────┴┐      ┌──────────┴┐     ┌───┴─┐           ┌───┴───┐       └┐   ┌─┴─┐       ┌───┴┐     ┌───┴─┐ ┌┴─┐               ┌───┴┐ ┌┴─────┐          ┌─┴─┐ ┌┴─────┐       ┌──────────┴─┐     ┌───┴┐   ┌┴┐ ┌─┴┐ ┌┴┐       ┌───┴┐     ┌───┴─┐ ┌┴─┐
+   ∧             -                   +                                        *             /  x      -      *    1  x      -   ∧                  *    ∧                                          +             ln      +           x      +           x     *     ∧           ∧       *        x   +   ∧       +    1     *    ln x  ∧               +    1 x      -          +  ln x      -       +           ln     *    x   x x ln 1 x x       +    1     *    ln x  ∧
┌───┴┐ ┌┴─────┐   ┌───┴┐        ┌─────────┴─────────────┐               ┌──────────┴─┐         ┌─┴┐    ┌───┴┐   ┌─┴─┐        ┌───┴┐ ┌┴─────┐          ┌─┴─┐ ┌┴─────┐                    ┌───────────────┴─┐           └┐  ┌───┴───┐          ┌───┴───┐           ┌─┴─┐  ┌┴┐         ┌┴─┐   ┌─┴────┐     ┌─┴┐ ┌┴┐     ┌┴─┐      ┌─┴─┐  └┐   ┌┴┐             ┌┴─┐       ┌───┴┐     ┌───┴┐ └┐    ┌───┴┐  ┌───┴───┐       └┐   ┌─┴─┐          └┐            ┌┴─┐      ┌─┴─┐  └┐   ┌┴┐
*    1 x      -   +    1        *                       *               +           ln         +  x    +    1   +   ∧        +    1 x      -          +  ln x      -                    +                 *            x  ∧       *          ∧       *           +   ∧  x 2         x  -   ∧      ∧     ln 1 x x     x  ∧      +   ∧   x   x x             x  ∧       +    2     *    1  x    +    1  ∧       *        x   +   ∧           x            x  ∧      +   ∧   x   x x
┌─┴─┐       ┌───┴┐ ┌┴─┐         ┌─┴─┐                  ┌──┴─┐         ┌───┴───┐       └┐     ┌───┴┐     ┌┴─┐    ┌─┴┐ ┌┴┐      ┌┴─┐       ┌───┴┐     ┌───┴┐ └┐    ┌───┴┐          ┌────────┴────────┐       ┌┴───┐          ┌┴─┐   ┌─┴────┐    ┌┴─┐   ┌─┴────┐    ┌─┴┐ ┌┴┐               ┌┴┐ ┌┴┐   ┌─┴┐    └┐             ┌┴┐   ┌─┴┐ ┌┴┐                        ┌┴┐     ┌┴─┐      ┌─┴─┐         ┌┴─┐    ┌┴─┐   ┌─┴────┐     ┌─┴┐ ┌┴┐                         ┌┴┐   ┌─┴┐ ┌┴┐
+   ∧       +    2 x  ∧         -   ∧                  *    ∧         ∧       *        x     *    1     x  ∧    ln 1 x x      x  ∧       +    2     *    1  x    +    1          +                 *       2    *          x  -   ∧      ∧    x  -   ∧      ∧    ln 1 x x               x 1 x x   +  2     x             x x   ln 1 x x                        x x     x  ∧      +   ∧         x  ∧    x  -   ∧      ∧     ln 1 x x                         x x   ln 1 x x
┌─┴┐ ┌┴┐     ┌┴─┐      ┌┴┐    ┌───┴┐ ┌┴─────┐          ┌─┴─┐ ┌┴─────┐  ┌┴─┐   ┌─┴────┐       ┌─┴─┐          ┌┴┐   └┐              ┌┴┐     ┌┴─┐      ┌─┴─┐         ┌┴─┐       ┌─────┴──┐          ┌───┴────┐     ┌─┴────┐       ┌┴┐ ┌┴┐   ┌─┴┐     ┌┴┐ ┌┴┐   ┌─┴┐   └┐                             ┌─┴┐                           └┐                                        ┌┴┐   ┌─┴┐ ┌┴┐          ┌┴┐     ┌┴┐ ┌┴┐   ┌─┴┐    └┐                                     └┐
ln 1 x x     x  ∧      x x    +    2 x      -          +  ln x      -  x  -   ∧      ∧       +   ∧          x x    x              x x     x  ∧      +   ∧         x  ∧       *        *          *        ∧     ∧      /       x 1 x x   +  2     x 1 x x   +  2    x                             ln 1                            x                                        x x   ln 1 x x          x x     x 1 x x   +  2     x                                      x
└┐             ┌┴┐           ┌┴─┐       ┌───┴┐     ┌───┴┐ └┐    ┌───┴┐   ┌┴┐ ┌┴┐   ┌─┴┐    ┌─┴┐ ┌┴┐                                         ┌┴┐   ┌─┴┐ ┌┴┐          ┌┴┐    ┌─┴─┐    ┌─┴─┐      ┌─┴─┐    ┌─┴┐   ┌┴┐   ┌─┴┐              ┌─┴┐               ┌─┴┐                                    └┐                                                                             └┐                                ┌─┴┐
x             x x           x  ∧       +    3     *    1  x    +    2   x 1 x x   +  2    ln 1 x x                                         x x   ln 1 x x          x x    -   ∧    ln  ∧      +   ∧    +  2   x x   +  x              ln 1               ln 1                                     x                                                                              x                                ln 1
┌┴┐     ┌┴─┐      ┌─┴─┐         ┌┴─┐              ┌─┴┐      └┐                                                     └┐                      ┌┴┐ ┌┴─┐  └┐ ┌┴─┐  ┌─┴┐ ┌┴┐ ┌─┴┐         ┌─┴┐                └┐                 └┐                                                                                                                                                       └┐
x x     x  ∧      +   ∧         x  ∧              ln 1       x                                                      x                      x 1 x  -   x x  -  ln 1 x x ln 1         ln 1                 x                  x                                                                                                                                                        x
┌┴┐   ┌─┴┐ ┌┴┐          ┌┴┐             └┐                                                                                             ┌┴┐      ┌┴┐ └┐       └┐           └┐
x x   ln 1 x x          x x              x                                                                                             x 2      x 1  x        x            x
└┐
x
References:

Knuth, D.E.,  "The Art of Computer Programming" Vol 1: "Fundamental Algorithms",
Ch 2.3.2.

Kahrimanian, H.G.,  "Analytical differentiation by a digital computer", master's
thesis, Temple University, May 1953.

"Symposium on Automatic Programming" (Washington, D.C: Office of Naval Research,
May 1954), 6-14.