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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-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 n
(make-exponentiation u (make-sum n -1)))
(deriv u var))))
(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)