I'm currently following the MIT Single Variable lectures online and the professor states that the binomial theorem for the expansion
$(x + \Delta x)^{n} = x^{n} + nx^{n-1}\Delta x + O((\Delta x)^{2})$
How is this derived, and what does the big O term of the expansion represent in terms of the binomial theorem. Just to put this in context, this expression was used when computing the derivative of $x^{n}$ using the limit definition.
As njguliyev writes, it means that the quantity $$\frac{(x+\Delta x)^n - (x^n + nx^{n-1} \Delta x)}{(\Delta x)^2}$$ is bounded, for sufficiently small $\Delta x$. That is, there exists an $M<\infty$ and an $\epsilon > 0$ such that for all $|\Delta x|<\epsilon$, we have $$|(x+\Delta x)^n - (x^n + nx^{n-1} \Delta x)| < M\,|(\Delta x)^2|.$$
The professor might have specified $\Delta x\to 0$ explicitly; if not, it was supposed to be implicit from the context.
Aside: Note that if $\Delta x$ is allowed to take arbitrarily large values, then the inequality cannot hold for any $M$. On the contrary, as $\Delta x\to\infty$ we get a different formula: $$(x+\Delta x)^n=(\Delta x)^n+nx(\Delta x)^{n-1}+O\left((\Delta x)^{n-2}\right).$$
