#lang sicp

(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 (exponentiation? x)
  (and (pair? x) (eq? (car x) '**)))
(define (base x) (cadr x))
(define (exponent x) (caddr x))
(define (make-exponentiation a b) 
  (cond ((=number? b 0) 1)
        ((=number? b 1) a)
        ((=number? a 0) 0)
        (else (list '** a b))))


 ;(list '^ a b))

(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))))
;; (1+a)*b*b
(deriv (make-product (make-product (make-sum 1 'a) 'b) 'b) 'b)
;; (+ (* (+ 1 a) b) (* (+ 1 a) b))
;; = 2(1+a)b. Correct!

(deriv (make-exponentiation 'x 3) 'x)
(deriv (make-exponentiation 'x 2) 'x)
(deriv (make-exponentiation 'x 1) 'x)

;;d((y*(x+2))**2)/dx = 2y**2 * (x+2)
(deriv (make-exponentiation (make-product 'y (make-sum 'x 2)) 2) 'x)