1. The problem is to find the derivative of the function $f(x) = 4x^3 + x^2 - x + 5$ using the limit definition of the derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 2. First, compute $f(x+h)$ by substituting $x+h$ into the function: $$f(x+h) = 4(x+h)^3 + (x+h)^2 - (x+h) + 5$$ 3. Expand each term: $$4(x+h)^3 = 4(x^3 + 3x^2h + 3xh^2 + h^3) = 4x^3 + 12x^2h + 12xh^2 + 4h^3$$ $$(x+h)^2 = x^2 + 2xh + h^2$$ $$-(x+h) = -x - h$$ 4. Combine all terms: $$f(x+h) = 4x^3 + 12x^2h + 12xh^2 + 4h^3 + x^2 + 2xh + h^2 - x - h + 5$$ 5. Now compute the difference $f(x+h) - f(x)$: $$f(x+h) - f(x) = (4x^3 + 12x^2h + 12xh^2 + 4h^3 + x^2 + 2xh + h^2 - x - h + 5) - (4x^3 + x^2 - x + 5)$$ Simplify by canceling terms: $$= 12x^2h + 12xh^2 + 4h^3 + 2xh + h^2 - h$$ 6. Divide by $h$: $$\frac{f(x+h) - f(x)}{h} = 12x^2 + 12xh + 4h^2 + 2x + h - 1$$ 7. Take the limit as $h \to 0$: $$f'(x) = \lim_{h \to 0} (12x^2 + 12xh + 4h^2 + 2x + h - 1) = 12x^2 + 2x - 1$$ 8. Therefore, the derivative of the function is: $$\boxed{f'(x) = 12x^2 + 2x - 1}$$