$f(x) = x^n$ $ \frac{df}{dx}=nx^{n-1} $ Use [[Binomial Expansion|binomial expansion]] $ \begin{align} \frac{df}{dx}&\approx\frac{1}{\Delta x}\left[\left(x+\Delta x\right)^n-x^n\right] \\ &\approx\frac{1}{\Delta x}\left[\cancel{x^n}+nx^{n-1}\Delta x+\frac{n(n-1)}{2}x^{n-2}\Delta x^2+O(\Delta x^3)-\cancel{x^n}\right] \\ &\approx\frac{1}{\Delta x}\left[nx^{n-1}\Delta x+\frac{n(n-1)}{2}x^{n-2}\Delta x^2+O(\Delta x^3)\right] \\ &\approx \boxed{nx^{n-1}+\cancel{O(\Delta x)}} \end{align} $