Extending basic differentiator to handle more kinds of expressions Exercise 2.56.
  
  Show how to extend the
  basic differentiator to handle more
  kinds of expressions. For instance,
  implement the differentiation rule
  
  $$ \frac{d(u^n)}{dr} = nu^{n-1}(\frac{du}{dr})$$
  
  by adding a new clause to the deriv
  program and defining appropriate
  procedures exponentiation?, base,
  exponent, and make-exponentiation.
  (You may use the symbol ** to denote
  exponentiation.) Build in the rules
  that anything raised to the power 0 is
  1 and anything raised to the power 1
  is the thing itself.


From the book:

(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))))


I added:

        ((exponentiation? exp)
         (make-product (exponent exp)
                       (make-exponentiation (base exp) (- (exponent exp) 1))))


Book:

        (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 (=number? exp num)
  (and (number? exp) (= exp num)))

(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 (augend s) (caddr s))

(define (product? x)
  (and (pair? x) (eq? (car x) '*)))

(define (multiplier p) (cadr p))

(define (multiplicand p) (caddr p))


I added:

(define (exponentiation? x)
  (and (pair? x) (eq? (car x) '**)))
(define (base s) (cadr s))
(define (exponent s) (caddr s))
(define (make-exponentiation m1 m2)
  (cond ((=number? m2 0) 1)
        ((= m2 1) m1)
        (else (list '** m1 m2))))


Can this be improved?
Your definitions are correct except for one minor detail.  The cond clause:

((= m2 1) m1)


should be:

((=number? m2 1) m1)


since one needs to test first whether m2 is a number.

Stylistically, I would prefer to define base and exponent as:

(define base cadr)
(define exponent caddr)


although there is certainly nothing wrong with your definitions.