Code

#lang racket

(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) ; dn/d(exp) = 0
         (let ((u (base exp))
               (n (exponent exp)))
           (make-product (make-product
                          n
                          (make-exponentiation u (make-sum n -1)))
                         (deriv u var))))
        (else
         (error "unknown expression type -- DERIV" exp))))

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

(define (base s) (cadr s))

(define (exponent s) (caddr s))

(define (make-exponentiation base exp)
  (cond ((=number? exp 0) 1)
        ((=number? exp 1) base)
        ((=number? base 1) 1)
        ((and (number? base) (number? exp)) (expt base exp))
        (else (list '** base exp))))

;;; representing algebraic expressions

(define (variable? x) (symbol? x))

(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

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

; tests

(deriv '(+ (* a (** x 2)) (* b x) c) 'x)
(deriv '(* a (* (** x 4) (** y 4))) 'x)
(deriv '(* 4 (** (+ x y) 3)) 'x)

Output

'(+ (* a (* 2 x)) b)
'(* a (* (* 4 (** x 3)) (** y 4)))
'(* 4 (* 3 (** (+ x y) 2)))