2.57

练习 2.57 请扩充求导程序，使之能处理任意项（两项或者更多项）的和与乘积。这样，上面最后一个例子就可以表示为：

(deriv '(* x y (+ x 3)) 'x)

设法通过只修改和与乘积的表示，而完全不修改过程 deriv 的方式完成这一扩充。例如，让一个和式的addend是它的第一项，而其augend是和式中的其余项。

先复用一下前面的代码：

(define (=number? v n)
    (if (number? v)
        (= v n)
        #f
    )
)

(define (error msg v)
    msg
)

(define variable? symbol?)
(define (same-variable? v1 v2)
    (and (variable? v1) (variable? v2) (eq? v1 v2))
)
(define (make-sum a1 a2) 
    (cond 
        ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list '+ a1 a2))
    )
)
(define (make-product m1 m2) 
    (cond
        ((=number? m1 0) 0)
        ((=number? m2 0) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        (else (list '* m1 m2))
    )
)
(define (sum? x) (and (pair? x) (eq? (car x) '+)))
(define addend cadr)
(define augend caddr)
(define (product? x) (and (pair? x) (eq? (car x) '*)))
(define multiplier cadr)
(define multiplicand caddr)

(define (deriv exp var)
    (cond 
        ((number? exp) 0)
        ((variable? exp)
            (if (same-variable? exp var) 1 0)
        )
        ((sum? exp)
            (make-sum
                (deriv (addend exp) var)
                (deriv (augend exp) var)
            )
        )
        ((product? exp)
            (make-product 
                (multiplier exp)
                (deriv (multiplicand exp) var)
            )
            (make-product
                (deriv (multiplier exp) var)
                (multiplicand exp)
            )
        )
        ((and (pair? exp) (= 1 (len exp)))
            (deriv (car exp) var)
        )
        (else
            (error exp)
        )
    )
)

; 期待返回 1
(deriv '(+ x 3) 'x)

; 期待得到 3x^2 的结果
(deriv '(** x 3) 'x)

定义 exponentiation?：

(define (exponentiation? exp)
    (and (pair? exp) (eq? (car exp) '**))
)

; 期待返回 true
(exponentiation? '(** x 3))

定义 base：

(define base cadr)

; 期待返回 x
(base '(** x 3))

定义 exponent：

(define exponent caddr)

; 期待返回 3
(exponent '(** x 3))

定义 make-exponentiation：

(define (make-exponentiation m1 m2) (list '** m1 m2))

; 期待返回 (** x 3)
(make-exponentiation 'x 3)

增加一个子句后的 deriv：

(define (deriv exp var)
    (cond 
        ((number? exp) 0)
        ((variable? exp)
            (if (same-variable? exp var) 1 0)
        )
        ((sum? exp)
            (make-sum
                (deriv (addend exp) var)
                (deriv (augend exp) var)
            )
        )
        ((product? exp)
            (make-sum

                (make-product 
                    (multiplier exp)
                    (deriv (multiplicand exp) var)
                )

                (make-product
                    (deriv (multiplier exp) var)
                    (multiplicand exp)
                )
            )
        )
        ((exponentiation? exp)
            (cond
                ((= (exponent exp) 0) 0)
                ((= (exponent exp) 1) 1)
                (else
                    (make-product
                        (exponent exp)
                        (make-exponentiation 
                            (base exp) 
                            (- (exponent exp) 1)
                        )
                    )
                )
            )
        )
        ((and (pair? exp) (= 1 (length exp)))
            (deriv (car exp) var)
        )
        (else
            (error "unknown expression type -- DERIV" exp)
        )
    )
)

(deriv '(** x 0) 'x)

(define addend cadr)
(define (augend exp)
    (if (= 1 (length (cddr exp)))
        (caddr exp)
        (cons '+ (cddr exp))
    )
)
(define multiplier cadr)
(define (multiplicand exp)
    (if (= 1 (length (cddr exp)))
        (caddr exp)
        (cons '* (cddr exp))
    )
)

; 期待得到 2xy + 3y 的结果
(deriv '(* x y (+ x 3)) 'x)