Applying the basic definition of derivative and binomial expansion gives

$$\begin{align*} \frac{d f(x)}{d x} =&\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\\ =&\lim_{\Delta x\to 0}\frac{(x+\Delta x)^n-x^n}{\Delta x}\\ =&\lim_{\Delta x\to 0}\frac{\left(x^n + nx^{n-1}\Delta x +\binom{n}{2}x^{n-1}\left(\Delta x\right)^2 +\cdots \right) - x^n}{\Delta x}\\ =&\lim_{\Delta x\to 0}\left(nx^{n-1}+\binom{n}{2}x^{n-2}\Delta x + \cdots\right) \end{align*}$$

Noting that, in the bracket, all the terms except the first one is a product of some powers of $\Delta x$ and $x$. These products should apporach $0$ when $\Delta x$ approaches $0$. The first term is independent of $\Delta x$. Therefore, we conclude

$$\boxed{\frac{d}{d x} x^n=nx^{n-1}}$$