#lang racket

(define (variable? x) (symbol? x))
(define (same-variable? x y) (and (variable? x) (variable? y) (eq? x y)))

(define (=number? x num)
  (and (number? x) (= x num)))
(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 (make-exponent e p)
  (cond ((=number? p 0) 1)
        ((=number? p 1) e)
        (else (list '** e p))))

(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 (exponent? x)
  (and (pair? x) (eq? (car x) '**)))
(define (base expo)
  (cadr expo))
(define (exponent expo)
  (caddr expo))

(define e '(+ a b))
(define v1 'a)
(define v2 'b)
(define m1 'a)
(define m2 'b)
(define a1 'a)
(define a2 'b)

(variable? e)
(same-variable? v1 v2)
(sum? e)
(addend e)
(augend e)
(make-sum a1 a2)
(product? e)
(multiplier e)
(multiplicand e)
(make-product m1 m2)

(define (deriv expr var)
  (cond ((number? expr) 0)
        ((variable? expr)
         (if (same-variable? expr var) 1 0))
        ((sum? expr)
         (make-sum (deriv (addend expr) var)
                   (deriv (augend expr) var)))
        ((product? expr)
         (make-sum
            (make-product 
              (multiplier expr)
              (deriv (multiplicand expr) var))
            (make-product 
              (deriv (multiplier expr) var)
              (multiplicand expr))))
        ((exponent? expr)
         (make-product
           (deriv (base expr) var)
           (make-product
             (exponent expr)
             (make-exponent (base expr) (- (exponent expr) 1)))))
        (else
          (error "unkown expression type -- DERIV" expr))))

(deriv '(+ x 3) 'x)
(deriv '(* x y) 'x)
(deriv '(* (* x y) (+ x 3)) 'x)

(deriv '(** (* 10 a) 3) 'a)
(deriv '(** a 2) 'a)