Problem with derivative of x^n

Question

I have a problem regarding the proof of the derivative of x^n using first principles.

Here's my proof.

D is for delta

y = x^n

y + Dy =(x+Dx) ^n

So Dy = (x+Dx) ^n - x^n

We can factor this as

(x +Dx -x) (x^(n-1)+....+Dx^(n-1)) = (Dx) (x^(n-1)+....+Dx^(n-1))

Dy/Dx = (x^(n-1)+....+Dx^(n-1))

When we limit Dx->0

dy/dx = x^(n-1)

But it should be nx^(n-1).

What am I missing?

Answer 1

In your notation it should be $$Dy=n(Dx)x^{n-1}+\frac{n(n-1)}{2}(Dx)^2x^{n-2}+...$$

Answer 2

You have not done the factorization correctly. $a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})$.

Answer 3

First principle means establishing $$\lim_{h\rightarrow 0}\frac{(x+h)^n-x^n}{h} = nx^{n-1}.$$ Here by the binomial theorem, $(x+h)^n = x^n + nhx^{n-1} + h^2z$ for some expression $z$. Then the limit gives $nx^{n-1}$ as claimed.