(define (decr x) (- x 1))

(define (deriv E)
    (cond
        ((number? E) 0) ; works
        ((symbol? E) 1) ; works
        ((eqv? (car E) '+) (list '+ (deriv (cadr E)) (deriv (caddr E)))) ; works
        ((eqv? (car E) '-) (list '- (deriv (cadr E)) (deriv (caddr E)))) ; works
        ((eqv? (car E) '*) (list '+ (list '* (cadr E) (deriv (caddr E))) (list '* (deriv (cadr E)) (caddr E)))) ; works
        ;((eqv? (car E) '^) (list '* (caddr E) (list '^ (cadr E) (decr (caddr E))) (deriv (cadr E)))) ; works
        ((eqv? (car E) '^) (list '* (list '^ (cadr E) (decr (caddr E))) (list '* (caddr E) (deriv (cadr E))))) ; works better
    )
)


; all cases of simplify are tested and working

(define (simplify E)
    (cond
        ((or (number? E) (symbol? E)) E)

    (else (let (
        (one (simplify (cadr E)))
        (two (simplify (caddr E)))
        )
        (cond
            ((and (eqv? (car E) '+) (eqv? two 0))
                one)
            ((and (eqv? (car E) '+) (eqv? one 0))
                two)
            ((and (eqv? (car E) '-) (eqv? two 0))
                one)
            ((and (eqv? (car E) '*) (or (eqv? one 0) (eqv? two 0)))
                0)
            ((and (eqv? (car E) '*) (eqv? one 1))
                two)
            ((and (eqv? (car E) '*) (eqv? two 1))
                one)
            ((and (eqv? (car E) '^) (eqv? two 1))
                one)
            ((and (eqv? (car E) '^) (eqv? one 1))
                1)
            ((and (eqv? (car E) '+) (and (number? one) (number? two)))
                (+ one two))
            ((and (eqv? (car E) '-) (and (number? one) (number? two)))
                (- one two))
            ((and (eqv? (car E) '*) (and (number? one) (number? two)))
                (* one two))
            ((and (eqv? (car E) '^) (and (number? one) (number? two)))
                (expt one two))
            (else E) ; else it gives up
        )
            )
          )
)
  )

(define (derivative E)
  (simplify (deriv E)))

(display (simplify (deriv '(+ (* a 1) (* 1 b)))))