1. **State the problem:** We need to simplify the difference quotient for the function $f(x) = \sqrt{15x}$, which is given by $$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$$ 2. **Write the expressions for $f(x+h)$ and $f(x)$:** $$f(x+h) = \sqrt{15(x+h)} = \sqrt{15x + 15h}$$ $$f(x) = \sqrt{15x}$$ 3. **Substitute into the difference quotient:** $$\frac{\sqrt{15x + 15h} - \sqrt{15x}}{h}$$ 4. **Rationalize the numerator to simplify:** Multiply numerator and denominator by the conjugate of the numerator: $$\frac{\sqrt{15x + 15h} - \sqrt{15x}}{h} \times \frac{\sqrt{15x + 15h} + \sqrt{15x}}{\sqrt{15x + 15h} + \sqrt{15x}} = \frac{(15x + 15h) - 15x}{h(\sqrt{15x + 15h} + \sqrt{15x})}$$ 5. **Simplify the numerator:** $$\frac{\cancel{15x} + 15h - \cancel{15x}}{h(\sqrt{15x + 15h} + \sqrt{15x})} = \frac{15h}{h(\sqrt{15x + 15h} + \sqrt{15x})}$$ 6. **Cancel $h$ in numerator and denominator:** $$\frac{15\cancel{h}}{\cancel{h}(\sqrt{15x + 15h} + \sqrt{15x})} = \frac{15}{\sqrt{15x + 15h} + \sqrt{15x}}$$ 7. **Final simplified form:** $$\boxed{\frac{15}{\sqrt{15x + 15h} + \sqrt{15x}}}$$ This is the simplified form of the difference quotient for $f(x) = \sqrt{15x}$.