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 `(load "displaylib.scm")` `(title "Exercise 2.56")` `(print "` ` Show how to extend the basic differentiator to handle more kinds of expressions. For instance, implement the differentiation rule` ` d(u^n) du` ` ------ = nu^{n-1} ----` ` dx dx` ` by adding a new clause to the deriv program and defining appropriate procedures exponentiation?, base, exponent, and make-exponentiation. (You may use the symbol ** to denote exponentiation.) Build in the rules that anything raised to the power 0 is 1 and anything raised to the power 1 is the thing itself. ` ` ")` ``` ``` `; -- 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))))` ``` ``` `(define (sum? x)` ` (and (pair? x) (eq? (car x) '+)))` `(define (addend s) (cadr s))` `(define (augend s) (caddr s))` ``` ``` `(define (product? x)` ` (and (pair? x) (eq? (car x) '*)))` `(define (multiplier p) (cadr p))` `(define (multiplicand p) (caddr p))` ``` ``` `; ---- END of given library -----` ``` ``` ``` ``` `; 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))))` ``` ``` `(display "(deriv '(+ x 3) 'x)")(newline)` `(display (deriv '(+ x 3) 'x))(newline)` `(display "(deriv '(* x y) 'x)")(newline)` `(display (deriv '(* x y) 'x))(newline)` `(display "(deriv '(* (* x y) (+ x 3)) 'x)")(newline)` `(display (deriv '(* (* x y) (+ x 3)) 'x))(newline)` `(display "(deriv '(** x y) 'x)")(newline)` `(display (deriv '(** x y) 'x))(newline)` `(display "(deriv '(** x 2) 'x)")(newline)` `(display (deriv '(** x 2) 'x))(newline)` `(display "(deriv '(** x (** x 2)) 'x)")(newline)` ```(display (deriv '(** x (** x 2)) 'x))(newline) ``` ``` ```