If f(x) = (√x)-8)/(√x)+8)

. Then find f'(7).

' (x) =

f'(7) =

Given \( f(x) = \frac{\sqrt{x} - 8}{\sqrt{x} + 8} \), we want to find \( f'(x) \) and then evaluate \( f'(7) \).

### Step 1: Rewrite the function for easier differentiation
Let \( u(x) = \sqrt{x} \), which simplifies the function to:
\[
f(x) = \frac{u(x) - 8}{u(x) + 8}
\]
Now apply the quotient rule for differentiation.

### Step 2: Apply the quotient rule
The quotient rule states:
\[
f'(x) = \frac{(g(x)h'(x) - h(x)g'(x))}{(h(x))^2}
\]
For \( f(x) = \frac{u(x) - 8}{u(x) + 8} \):
- \( g(x) = u(x) - 8 \)
- \( h(x) = u(x) + 8 \)
- \( u(x) = \sqrt{x} \) so \( u'(x) = \frac{1}{2\sqrt{x}} \)

Now, let's differentiate:
\[
f'(x) = \frac{(u'(x) \cdot (u(x) + 8) - u'(x) \cdot (u(x) - 8))}{(u(x) + 8)^2}
\]

### Step 3: Simplify the derivative expression
Substituting \( u'(x) = \frac{1}{2\sqrt{x}} \) and \( u(x) = \sqrt{x} \):
\[
f'(x) = \frac{\frac{1}{2\sqrt{x}} \cdot (\sqrt{x} + 8) - \frac{1}{2\sqrt{x}} \cdot (\sqrt{x} - 8)}{(\sqrt{x} + 8)^2}
\]
Simplifying the numerator:
\[
f'(x) = \frac{\frac{1}{2\sqrt{x}} \left( (\sqrt{x} + 8) - (\sqrt{x} - 8) \right)}{(\sqrt{x} + 8)^2}
\]
\[
f'(x) = \frac{\frac{1}{2\sqrt{x}} \cdot (16)}{(\sqrt{x} + 8)^2}
\]
\[
f'(x) = \frac{8}{\sqrt{x} \cdot (\sqrt{x} + 8)^2}
\]

### Step 4: Evaluate at \( x = 7 \)
Now substitute \( x = 7 \) into the derivative:
\[
f'(7) = \frac{8}{\sqrt{7} \cdot (\sqrt{7} + 8)^2}
\]

Simplifying further:
- \( \sqrt{7} \approx 2.6458 \)
- \( \sqrt{7} + 8 \approx 10.6458 \)
- \( (10.6458)^2 \approx 113.3457 \)

Now substitute these values:
\[
f'(7) = \frac{8}{2.6458 \cdot 113.3457} \approx \frac{8}{299.74} \approx 0.0267
\]

Thus, \( f'(7) \approx 0.0267 \).

### Final Answer:
\[
f'(7) \approx 0.0267
\]

---

Would you like further details or clarifications?

Here are 5 related questions for practice:
1. How do you apply the chain rule when differentiating functions involving square roots?
2. How would you simplify the function if \( f(x) = \frac{\sqrt{x} + 8}{\sqrt{x} - 8} \)?
3. What happens to \( f'(x) \) as \( x \) approaches 0 in this case?
4. Can you find the second derivative \( f''(x) \) of the function \( f(x) \)?
5. How do you evaluate limits of functions like \( f(x) \) as \( x \to \infty \)?

**Tip:** When differentiating fractions, the quotient rule can be challenging. Practice the rule step-by-step to ensure accuracy in applying it.

If f(x) = (√x - 8)/(√x + 8), find f'(7).

Explore the step-by-step process of differentiating the function f(x) = (√x - 8)/(√x + 8) using the quotient rule and evaluating it at f'(7). This problem covers key calculus concepts like differentiation, the quotient rule, and working with square roots. Suitable for advanced high school students in Grades 11-12.

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