1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $y = x^n$ using the first principle of derivatives. 2. **Recall the first principle of derivatives:** The derivative of a function $f(x)$ at a point $x$ is given by $$\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Apply the formula to $y = x^n$:** $$\frac{dy}{dx} = \lim_{h \to 0} \frac{(x+h)^n - x^n}{h}$$ 4. **Expand $(x+h)^n$ using the binomial theorem:** $$(x+h)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} h^k = x^n + n x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \cdots + h^n$$ 5. **Substitute the expansion back:** $$\frac{dy}{dx} = \lim_{h \to 0} \frac{x^n + n x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \cdots + h^n - x^n}{h}$$ 6. **Simplify numerator by canceling $x^n$:** $$\frac{dy}{dx} = \lim_{h \to 0} \frac{n x^{n-1} h + \binom{n}{2} x^{n-2} h^2 + \cdots + h^n}{h}$$ 7. **Factor out $h$ from numerator:** $$\frac{dy}{dx} = \lim_{h \to 0} \frac{h \left(n x^{n-1} + \binom{n}{2} x^{n-2} h + \cdots + h^{n-1}\right)}{h}$$ 8. **Cancel $h$ in numerator and denominator:** $$\frac{dy}{dx} = \lim_{h \to 0} \cancel{\frac{h}{h}} \left(n x^{n-1} + \binom{n}{2} x^{n-2} h + \cdots + h^{n-1}\right) = \lim_{h \to 0} \left(n x^{n-1} + \binom{n}{2} x^{n-2} h + \cdots + h^{n-1}\right)$$ 9. **Take the limit as $h \to 0$:** All terms with $h$ vanish, leaving $$\frac{dy}{dx} = n x^{n-1}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = n x^{n-1}}$$