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 `(load "displaylib.scm")` `(title "Exercise 2.58 a")` `(print "` ` Suppose we want to modify the differentiation program so that it works with ordinary mathematical notation, in which + and * are infix rather than prefix operators. Since the differentiation program is defined in terms of abstract data, we can modify it to work with different representations of expressions solely by changing the predicates, selectors, and constructors that define the representation of the algebraic expressions on which the differentiator is to operate.` ` a. Show how to do this in order to differentiate algebraic expressions presented in infix form, such as (x + (3 * (x + (y + 2)))). To simplify the task, assume that + and * always take two arguments and that expressions are fully parenthesized.` ` ")` ``` ``` `; -- Given library` ``` ``` `(define (variable? x) (symbol? x))` `(define (same-variable? v1 v2)` ` (and (variable? v1) (variable? v2) (eq? v1 v2)))` ``` ``` `; Naive` `; (define (make-sum a1 a2) (list '+ a1 a2))` `; (define (make-product m1 m2) (list '* m1 m2))` ``` ``` `(define (=number? x v)` ` (and (number? x) (= x v)))` `(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 ((or (=number? m1 0) (=number? m2 0)) 0)` ` ((=number? m1 1) m2)` ` ((=number? m2 1) m1)` ` ((and (number? m1) (number? m2)) (* m1 m2))` ` (else (list '* m1 m2))))` ``` ``` ``` ``` `; Added exponentation case` `(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)` ` (let ((u (base exp))` ` (n (exponent exp)))` ` (make-product` ` (make-product n` ` (make-exponentiation u (make-sum n -1)))` ` (deriv u var))))` ` (else` ` (error "unknown expression type -- DERIV" exp))))` ``` ``` `(define (exponentiation? x) (and (pair? x) (eq? (car x) '**)))` `(define (base e) (cadr e))` `(define (exponent e) (caddr e))` `(define (make-exponentiation u n)` ` (cond ((=number? n 0) 1)` ` ((=number? n 1) u)` ` ((and (number? u) (number? n)) (pow u n))` ` (else (list '** u n))))` ``` ``` `; ---- END of given library -----` ``` ``` `; -- Amazingly easy to change representation of sum and product to infix` `(define (sum? x)` ` (and (pair? x) (eq? (cadr x) '+)))` `(define (addend s) (car s))` `(define (augend s) (fold make-sum (caddr s) (cdddr s)))` ``` ``` `(define (product? x)` ` (and (pair? x) (eq? (cadr x) '*)))` `(define (multiplier p) (car p))` `(define (multiplicand p) (fold make-product (caddr p) (cdddr p)))` ``` ``` `(display "(deriv '(x + 3) 'x)")(newline)` `(display (deriv '(x + 3) 'x))(newline)` `(display "(deriv '(x + (3 * (x + (y + 2)))) 'x)")(newline)` ```(display (deriv '(x + (3 * (x + (y + 2)))) 'x))(newline) ``` ``` ```