1. **Problem:** Find the derivative of the function $f(x) = 3x^2 - 5x$ 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:** Calculate $f(x+h)$: $$f(x+h) = 3(x+h)^2 - 5(x+h) = 3(x^2 + 2xh + h^2) - 5x - 5h = 3x^2 + 6xh + 3h^2 - 5x - 5h$$ 4. **Step 2:** Substitute into the difference quotient: $$\frac{f(x+h) - f(x)}{h} = \frac{(3x^2 + 6xh + 3h^2 - 5x - 5h) - (3x^2 - 5x)}{h} = \frac{6xh + 3h^2 - 5h}{h}$$ 5. **Step 3:** Simplify by canceling $h$: $$\frac{\cancel{h}(6x + 3h - 5)}{\cancel{h}} = 6x + 3h - 5$$ 6. **Step 4:** Take the limit as $h \to 0$: $$f'(x) = \lim_{h \to 0} (6x + 3h - 5) = 6x - 5$$ **Final answer:** $$f'(x) = 6x - 5$$