The problem asks to find the derivative $f'(x)$ of the function $f(x) = \frac{7 - x^2}{7 + x^2}$ using the Quotient Rule, and to also evaluate $f'(2)$.

The Quotient Rule states that for a function $f(x) = \frac{u(x)}{v(x)}$, the derivative is:

$f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$

Here, we can identify:

• $u(x) = 7 - x^2$
• $v(x) = 7 + x^2$

Now, let's differentiate both:

• $u'(x) = -2x$
• $v'(x) = 2x$

Applying the Quotient Rule:

$f'(x) = \frac{(7 + x^2)(-2x) - (7 - x^2)(2x)}{(7 + x^2)^2}$

Let's simplify this expression:

$f'(x) = \frac{-2x(7 + x^2) - 2x(7 - x^2)}{(7 + x^2)^2}$

$f'(x) = \frac{-2x(7 + x^2 + 7 - x^2)}{(7 + x^2)^2}$

$f'(x) = \frac{-2x(14)}{(7 + x^2)^2}$

$f'(x) = \frac{-28x}{(7 + x^2)^2}$

Finding $f'(2)$:

Now, substitute $x = 2$ into the derivative:

$f'(2) = \frac{-28(2)}{(7 + 2^2)^2} = \frac{-28(2)}{(7 + 4)^2} = \frac{-56}{11^2} = \frac{-56}{121}$

So, $f'(2) = \frac{-56}{121}$.

Let me know if you'd like more details, or if you have any questions!

Here are 5 related questions to expand on this topic:

1. How is the quotient rule derived from the product and chain rules?
2. What other methods can be used to differentiate rational functions besides the quotient rule?
3. How would the result change if the numerator and denominator functions were different polynomials?
4. How does the derivative behavior change as $x \to \infty$?
5. How can the quotient rule be applied to trigonometric functions?

Tip: Always ensure that the denominator is not zero when applying the quotient rule.