(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 (exponent exp)
                       (make-product (deriv (base exp) var)
                                     (make-exponentiation (base exp) (- (exponent exp) 1)))))
        (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 (sum? x)
  (and (pair? x) (eq? (car x) '+)))
(define (addend s) (cadr s))
(define (product? x)
  (and (pair? x) (eq? (car x) '*)))
(define (multiplier p) (cadr p))
(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define (exponentiation? x)
  (and (pair? x) (eq? (car x) '**)))
(define (base x) (cadr x))
(define (exponent x) (caddr x))
(define (make-exponentiation base exponent)
  (cond ((=number? exponent 0) 1)
        ((=number? exponent 1) base)
        ((and (number? exponent) (number? base)) (fast-exp base exponent))
        (else (list '** base exponent))))
(define (fast-expt b n)
  (cond ((= n 0) 1)
        ((even? n) (square (fast-expt b (/ n 2))))
        (else (* b (fast-expt b (- n 1))))))
(define (square x) (* x x))

;Solution 2.57
(define (augend s)
  (if (null? (cdddr s)) (caddr s)
      (cons '+ (cddr s))))
(define (multiplicand p)
  (if (null? (cdddr p)) (caddr p)
      (cons '* (cddr p))))