Code

```#lang racket

(define (deriv exp var)
(cond ((number? exp) 0)
((variable? exp)
(if (same-variable? exp var) 1 0))
((sum? exp)
(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 (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 (product? x)
(and (pair? x) (eq? (car x) '*)))

(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)))
```