Let's calculate the derivative of the function \( f(x) = x^n \) using the limit definition of the derivative:

\begin{align} f'(x_0) &= \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \\ &= \lim_{x \to x_0} \frac{x^n - x_0^n}{x - x_0}\end{align}

The numerator of the difference quotient is the difference of powers \( x^n - x_0^n \):

\[ x^n - x_0^n = (x - x_0)(x^{n-1} + x^{n-2} x_0 + \cdots + x_0^{n-1}) \]

Substituting this into the expression for the derivative and then simplifying:

\begin{align} f'(x_0) &= \lim_{x \to x_0} \frac{(x - x_0)(x^{n-1} + x^{n-2} x_0 + \cdots + x_0^{n-1})}{x - x_0} \\ &= \lim_{x \to x_0} \left(x^{n-1} + x^{n-2} x_0 + \cdots + x_0^{n-1}\right) \end{align}

As \( x \to x_0 \), all terms are evaluated at \( x_0 \):

\[ f'(x_0) = n x_0^{n-1} \]

Therefore, the derivative of the function \( f(x) = x^n \) is:

\[ f'(x) = n x^{n-1} \qquad \forall x \in \mathbb{R} \]