1. **Problem statement:** Find the derivative of the function $f(x) = 2x^2 - x + 5$ and then evaluate it at $x=3$. 2. **Formula and rules:** The derivative of a function $f(x)$, denoted $f'(x)$, is found by applying the power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ and the derivative of a constant is zero. 3. **Calculate the derivative:** $$f(x) = 2x^2 - x + 5$$ Apply the derivative term-by-term: $$f'(x) = \frac{d}{dx}(2x^2) - \frac{d}{dx}(x) + \frac{d}{dx}(5)$$ Using the power rule: $$f'(x) = 2 \cdot 2x^{2-1} - 1 \cdot x^{1-1} + 0 = 4x - 1$$ 4. **Evaluate the derivative at $x=3$:** $$f'(3) = 4 \cdot 3 - 1 = 12 - 1 = 11$$ **Final answer:** $$f'(3) = 11$$