Almost every book of differential calculus treats the proof of the fundamental formula
(d/(dx))(x^n) = nx^<n-1>
as follows :
(1) Using the formula
x^n_2 - x^n_1 = (x_2-x_1)(x^<n-1>_2+x^<n-2>_2x_1+・・・+x^<n-1>_1)
(without any proof of this formula), or
(2）Using the binomial formula.
The former method has a defeat, that is, the used formula has no proof in the book, generally.
The latter method is quite complete but it needs the binomial formula to be proved. For this purpose we have to teach 'permutation and combination' throughly, although it takes much time and it decreases ordiuary students' interest in mathematics and let them feel weary. To avoid these I advise to use the following metbod :
First derive the formulae of differentiation of the sum, the product, and the quotient of two functions and of the composite function.※ Then with the mathematical induction and these formulae the derivative of x^n would be obtained easily without the use of the binomial theorem.
※ If n is limited to be an integer, this is not necessary.