I just watched the first lecture of

MIT 18.01 Single Variable Calculus, Fall 2006, and in that video, the lecturer explained about the geometric interpretation of derivative. A derivative of a function

can be defined as:

The lecturer then used this to find the derivative of the function

, and this is where it starts to get confusing for me. Using the definition above, the derivate of the function

is equivalent to:

The part which confuses me is when the term

is evaluated using the

Binomial Theorem:

Why are the rest of the terms replaced by the Big-O Notation

? And why the term

specifically?

Thanks!