1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = 5x + \sqrt{3 - x}$$ using the definition of the derivative. 2. **Recall the definition of the derivative:** $$\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Apply the definition to our function:** $$\frac{dy}{dx} = \lim_{h \to 0} \frac{5(x+h) + \sqrt{3 - (x+h)} - \left(5x + \sqrt{3 - x}\right)}{h}$$ 4. **Simplify the numerator:** $$= \lim_{h \to 0} \frac{5x + 5h + \sqrt{3 - x - h} - 5x - \sqrt{3 - x}}{h}$$ $$= \lim_{h \to 0} \frac{5h + \sqrt{3 - x - h} - \sqrt{3 - x}}{h}$$ 5. **Separate the limit into two parts:** $$= \lim_{h \to 0} \frac{5h}{h} + \lim_{h \to 0} \frac{\sqrt{3 - x - h} - \sqrt{3 - x}}{h}$$ 6. **Simplify the first term:** $$\lim_{h \to 0} \frac{5h}{h} = \lim_{h \to 0} 5 = 5$$ 7. **Focus on the second term:** Use the conjugate to simplify: $$\frac{\sqrt{3 - x - h} - \sqrt{3 - x}}{h} \times \frac{\sqrt{3 - x - h} + \sqrt{3 - x}}{\sqrt{3 - x - h} + \sqrt{3 - x}}$$ 8. **Multiply numerator and denominator:** $$= \frac{(3 - x - h) - (3 - x)}{h \left(\sqrt{3 - x - h} + \sqrt{3 - x}\right)} = \frac{-h}{h \left(\sqrt{3 - x - h} + \sqrt{3 - x}\right)}$$ 9. **Cancel $h$ in numerator and denominator:** $$= \frac{\cancel{-h}}{\cancel{h} \left(\sqrt{3 - x - h} + \sqrt{3 - x}\right)} = \frac{-1}{\sqrt{3 - x - h} + \sqrt{3 - x}}$$ 10. **Take the limit as $h \to 0$:** $$\lim_{h \to 0} \frac{-1}{\sqrt{3 - x - h} + \sqrt{3 - x}} = \frac{-1}{\sqrt{3 - x} + \sqrt{3 - x}} = \frac{-1}{2\sqrt{3 - x}}$$ 11. **Combine both parts:** $$\frac{dy}{dx} = 5 - \frac{1}{2\sqrt{3 - x}}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = 5 - \frac{1}{2\sqrt{3 - x}}}$$