The image asks for the difference quotient of the function $f(x) = \sqrt{x + 6}$ in its simplest form.

The difference quotient is given by:

$\frac{f(x+h) - f(x)}{h}$

Step 1: Apply the function $f(x) = \sqrt{x+6}$

• $f(x+h) = \sqrt{x+h+6}$
• $f(x) = \sqrt{x+6}$

So, the difference quotient becomes:

$\frac{\sqrt{x+h+6} - \sqrt{x+6}}{h}$

Step 2: Simplify the expression

To simplify the expression, multiply both the numerator and the denominator by the conjugate of the numerator, which is $\sqrt{x+h+6} + \sqrt{x+6}$:

$\frac{\sqrt{x+h+6} - \sqrt{x+6}}{h} \cdot \frac{\sqrt{x+h+6} + \sqrt{x+6}}{\sqrt{x+h+6} + \sqrt{x+6}} = \frac{(\sqrt{x+h+6})^2 - (\sqrt{x+6})^2}{h(\sqrt{x+h+6} + \sqrt{x+6})}$

Step 3: Simplify further

The squares of the square roots cancel out:

$\frac{(x+h+6) - (x+6)}{h(\sqrt{x+h+6} + \sqrt{x+6})}$

Simplifying the numerator:

$\frac{x+h+6 - x - 6}{h(\sqrt{x+h+6} + \sqrt{x+6})} = \frac{h}{h(\sqrt{x+h+6} + \sqrt{x+6})}$

Canceling out the $h$'s:

$\frac{1}{\sqrt{x+h+6} + \sqrt{x+6}}$

Conclusion

The simplified form of the difference quotient is:

$\frac{1}{\sqrt{x+h+6} + \sqrt{x+6}}$

This matches option d in the image.

Would you like further details on any step?

Related Questions:

1. What is the purpose of multiplying by the conjugate?
2. How is the difference quotient used in finding derivatives?
3. What does the simplification of the numerator rely on?
4. Why does the expression involve square roots?
5. What happens to the difference quotient as $h \to 0$?

Tip: When simplifying expressions involving square roots, multiplying by the conjugate can help eliminate the square roots in the numerator.