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)
​x
 
Error: execute: unbound symbol: "deriv" []true

 
(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)
 
1

 
; 期待得到 3x^2 的结果
(deriv '(** x 3) 'x)
 
unknown expression type -- DERIV

定义 exponentiation?：

 
(define (exponentiation? exp)
  (and (pair? exp) (eq? (car exp) '**))
  )
​
; 期待返回 true
(exponentiation? '(** x 3))
 
true

定义 base：

 
(define base cadr)
​
; 期待返回 x
(base '(** x 3))
 
'x

定义 exponent：

 
(define exponent caddr)
​
; 期待返回 3
(exponent '(** x 3))
 
3

定义 make-exponentiation：

 
(define (make-exponentiation m1 m2) (list '** m1 m2))
​
; 期待返回 (** x 3)
(make-exponentiation 'x 3)
 
('** '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)
 
0

 
; 期待得到 3x^2 的结果
(deriv '(** x 3) 'x)
 
('* 3 ('** 'x 2))