Exercise 2.56. Show how to extend the basic differentiator to handle more kinds of expressions. For instance, implement the differentiation rule
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.
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| (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) | |
| (make-product | |
| (make-product | |
| (exponent exp) | |
| (make-exponentiation | |
| (base exp) | |
| (make-sum | |
| (exponent exp) | |
| -1))) | |
| (deriv (base exp) var))) | |
| (else | |
| (error "unknown expression type — DERIV" exp)))) | |
| (define (variable? x) (symbol? x)) | |
| (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) | |
| (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 (=number? exp num) | |
| (and (number? exp) (= exp num))) | |
| (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)) | |
| (define (base exp) (cadr exp)) | |
| (define (exponent exp) (caddr exp)) | |
| (define (make-exponentiation base exp) | |
| (cond | |
| ((=number? base 1) 1) | |
| ((=number? exp 1) base) | |
| ((=number? exp 0) 1) | |
| (else | |
| (list '** base exp)))) | |
| (define (exponentiation? exp) | |
| (and | |
| (pair? exp) | |
| (eq? (car exp) '**))) | |
| (deriv (make-exponentiation 'x 3) 'x) | |