hey everyone, I'm teaching myself calculus, because I felt it would be useful, even if I'm not going to use it in accounting. I am learning about the binomial theorem and differentiating negative indexes.

Here is the current problem I'm working on

1.) y=x^-2

2.) y+dy=(x+dx)^-2

3.) =x^-2(1+ dx/x)^-2

4.) =x^-2 [1-2dx/x + 2(2-1)/1*2 * (dx/x)^2 - etc. ]

5.) =x^-2 -2x^-3*dx + 3x^-4(dx)^2 -4x^-5(dx)^3 +etc...

Ignoring small units

6.) y+dy=x^-2 -2x^-3*dx

Subtracting original y=x^-2

7.) dy= -2x^-3*dx

8.) dy/dx= -2x^-3

the binomial theorem as it is explained in the book follows,

(a+b)^n=a^n + n(a^n-1b/1) + n(n-1)(a^n-2b^2/2) + n(n-1)(n-2)(a^n-3b/3) + etc....

I understand how one moves from Step 1 to 2, that being said how does (x+dx)^-2 become x^-2(1+dx/x)^-2? Secondly, could someone how one goes about applying the binomial theorem to a negative index?

any help would be greatly appreciated