1. **Problem statement:** Differentiate the function $f(x) = x^{100}$ using the binomial formula. 2. **Recall the binomial formula:** The binomial theorem states that for any integer $n$, $$ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $$ However, this formula is primarily used for expanding powers of sums, not directly for differentiation. 3. **Differentiation of $x^{100}$:** The function $x^{100}$ is a simple power function. The derivative of $x^n$ with respect to $x$ is given by the power rule: $$ \frac{d}{dx} x^n = n x^{n-1} $$ 4. **Apply the power rule:** $$ \frac{d}{dx} x^{100} = 100 x^{99} $$ 5. **Explanation:** The binomial formula is not necessary here because $x^{100}$ is not a sum raised to a power but a single term. The power rule is the most straightforward and appropriate method. **Final answer:** $$ \boxed{100 x^{99}} $$