1. **Stating the problem:** Find the derivative of the function $$f(x) = 2x^3 - 9x^2 + 12x$$ and then evaluate it at $$x = 2$$ to find $$f'(2)$$. 2. **Formula and rules:** The derivative of a function $$f(x)$$ with respect to $$x$$ is given by $$f'(x) = \frac{d}{dx}f(x)$$. For power functions, the derivative rule is: $$\frac{d}{dx} x^n = nx^{n-1}$$ The derivative of a sum/difference is the sum/difference of the derivatives. 3. **Find the derivative:** Given $$f(x) = 2x^3 - 9x^2 + 12x$$, Apply the derivative term-by-term: $$f'(x) = \frac{d}{dx}(2x^3) - \frac{d}{dx}(9x^2) + \frac{d}{dx}(12x)$$ Using the power rule: $$= 2 \cdot 3x^{3-1} - 9 \cdot 2x^{2-1} + 12 \cdot 1x^{1-1}$$ $$= 6x^2 - 18x + 12$$ 4. **Evaluate the derivative at $$x=2$$:** $$f'(2) = 6(2)^2 - 18(2) + 12$$ $$= 6 \cdot 4 - 36 + 12$$ $$= 24 - 36 + 12$$ $$= 0$$ **Final answer:** $$f'(2) = 0$$