(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))))