2.56

练习 2.56 请说明如何扩充基本求导规则，以便能够处理更多种类的表达式。例如，通过给程序deriv增加一个新子句，并以适当方式定义过程exponentiation?、base、exponent和make-exponentiation的方式，实现下述求导规则（你可以考虑用符号**表示乘幂）：

$$\frac{d(u^n)}{dx}=nu^{n-1}(\frac{du}{dx})$$

请将如下规则也构造到程序里：任何东西的0次幂都是1，而它们的1次幂都是其自身。

; 期待得到 3x^2 的结果
(deriv '(** x 3) 'x)

(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) (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)
            )
        )
        (else
            (error "unknown expression type -- DERIV" 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-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)
                        )
                    )
                )
            )
        )
        (else
            (error "unknown expression type -- DERIV" exp)
        )
    )
)

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

; 期待得到 3x^2 的结果
(deriv '(** x 3) 'x)