1. **Problem:** Determine $f'(x)$ given that $f(x) = 4x^3$ from first principles. 2. **Formula:** The derivative from first principles is given by: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Step 1:** Substitute $f(x) = 4x^3$ into the formula: $$f'(x) = \lim_{h \to 0} \frac{4(x+h)^3 - 4x^3}{h}$$ 4. **Step 2:** Expand $(x+h)^3$ using binomial expansion: $$(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$$ 5. **Step 3:** Substitute expansion back: $$f'(x) = \lim_{h \to 0} \frac{4(x^3 + 3x^2h + 3xh^2 + h^3) - 4x^3}{h}$$ 6. **Step 4:** Simplify numerator: $$= \lim_{h \to 0} \frac{4x^3 + 12x^2h + 12xh^2 + 4h^3 - 4x^3}{h}$$ $$= \lim_{h \to 0} \frac{12x^2h + 12xh^2 + 4h^3}{h}$$ 7. **Step 5:** Cancel $h$ in numerator and denominator: $$= \lim_{h \to 0} \frac{\cancel{h}(12x^2 + 12xh + 4h^2)}{\cancel{h}} = \lim_{h \to 0} (12x^2 + 12xh + 4h^2)$$ 8. **Step 6:** Take the limit as $h \to 0$: $$f'(x) = 12x^2 + 12x \cdot 0 + 4 \cdot 0^2 = 12x^2$$ **Final answer:** $$\boxed{f'(x) = 12x^2}$$