1. **State the problem:** Find the difference quotient for the function $f(x) = \sqrt{x} + 4$ and simplify it. 2. **Recall the difference quotient formula:** $$\frac{f(x+h) - f(x)}{h}$$ where $h \neq 0$. 3. **Calculate $f(x+h)$:** $$f(x+h) = \sqrt{x+h} + 4$$ 4. **Form the difference quotient numerator:** $$f(x+h) - f(x) = (\sqrt{x+h} + 4) - (\sqrt{x} + 4) = \sqrt{x+h} - \sqrt{x}$$ 5. **Write the difference quotient:** $$\frac{\sqrt{x+h} - \sqrt{x}}{h}$$ 6. **Simplify the numerator by rationalizing:** Multiply numerator and denominator by the conjugate $\sqrt{x+h} + \sqrt{x}$: $$\frac{\sqrt{x+h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}} = \frac{(\sqrt{x+h})^2 - (\sqrt{x})^2}{h(\sqrt{x+h} + \sqrt{x})} = \frac{(x+h) - x}{h(\sqrt{x+h} + \sqrt{x})}$$ 7. **Simplify the numerator:** $$\frac{h}{h(\sqrt{x+h} + \sqrt{x})}$$ 8. **Cancel $h$ in numerator and denominator:** $$\frac{\cancel{h}}{\cancel{h}(\sqrt{x+h} + \sqrt{x})} = \frac{1}{\sqrt{x+h} + \sqrt{x}}$$ **Final simplified difference quotient:** $$\frac{1}{\sqrt{x+h} + \sqrt{x}}$$