(define (cadr s) (car (cdr s)))
(define (caddr s) (car (cdr (cdr s))))

; derive returns the derivative of EXPR with respect to VAR
(define (derive expr var)
  (cond ((number? expr) 0)
        ((variable? expr) (if (same-variable? expr var) 1 0))
        ((sum? expr) (derive-sum expr var))
        ((product? expr) (derive-product expr var))
        ((exp? expr) (derive-exp expr var))
        (else 'Error)))

; Variables are represented as symbols
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

; Numbers are compared with =
(define (=number? expr num)
  (and (number? expr) (= expr num)))

; Sums are represented as lists that start with +.
(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 (sum? x)
  (and (list? x) (eq? (car x) '+)))
(define (first-operand s) (cadr s))
(define (second-operand s) (caddr s))

; Products are represented as lists that start with *.
(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 (product? x)
  (and (list? x) (eq? (car x) '*)))
; You can access the operands from the expressions with
; first-operand and second-operand
(define (first-operand p) (cadr p))
(define (second-operand p) (caddr p))

(define (derive-sum expr var)
  (make-sum (derive (first-operand expr) var) (derive (second-operand expr) var))
)

(define (derive-product expr var)
  (make-sum
   (make-product (derive (first-operand expr)  var) (second-operand expr) ) 
   (make-product (derive (second-operand expr) var) (first-operand expr) ) 
  )
)

; Exponentiations are represented as lists that start with ^.
(define (make-exp base exponent)
  (define (exp x a)
    (cond
    ( (or (= 1 x) (= 0 a)) 1)
    (else (* x (exp x (- a 1))))
    )
  )
  (cond
  ((=number? exponent 1) base)
  ((or (=number? exponent 0) (=number? base 1)) 1)
  ((and (number? base) (number? exponent)) (exp base exponent))
  (else (list '^ base exponent)))
)

(define (exp? exp)
  (and (list? exp) (eq? '^ (car exp)))
)

(define x^2 (make-exp 'x 2))
(define x^3 (make-exp 'x 3))

(define (derive-exp exp var)
  (make-product (second-operand exp) (make-exp (first-operand exp) (- (second-operand exp) 1)))
)