To evaluate $f'(x)$ for $f(x) = \frac{(5x^2 - 7)(x^2 - 6)}{x^2 + 6}$ at $x = 2$, we'll follow these steps:

1. Simplify the function if possible.
2. Differentiate the function.
3. Evaluate the derivative at $x = 2$.

Step 1: Simplify the function if possible

Given $f(x) = \frac{(5x^2 - 7)(x^2 - 6)}{x^2 + 6}$, it's not immediately obvious how to simplify it, so we proceed to differentiate it using the quotient rule.

Step 2: Differentiate the function

The quotient rule states that if you have a function $g(x) = \frac{u(x)}{v(x)}$, then the derivative $g'(x)$ is given by: $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Here, we let: $u(x) = (5x^2 - 7)(x^2 - 6)$ $v(x) = x^2 + 6$

Differentiate $u(x)$:

To find $u'(x)$, we use the product rule: $u(x) = (5x^2 - 7)(x^2 - 6)$ $u'(x) = (5x^2 - 7)'(x^2 - 6) + (5x^2 - 7)(x^2 - 6)'$

Calculate the derivatives: $(5x^2 - 7)' = 10x$ $(x^2 - 6)' = 2x$

Thus, $u'(x) = 10x(x^2 - 6) + (5x^2 - 7)2x$ $u'(x) = 10x^3 - 60x + 10x^3 - 14x$ $u'(x) = 20x^3 - 74x$

Differentiate $v(x)$:

$v(x) = x^2 + 6$ $v'(x) = 2x$

Apply the quotient rule:

$f'(x) = \frac{(20x^3 - 74x)(x^2 + 6) - (5x^2 - 7)(x^2 - 6)2x}{(x^2 + 6)^2}$

Simplify the numerator: $(20x^3 - 74x)(x^2 + 6) = 20x^5 + 120x^3 - 74x^3 - 444x$ $= 20x^5 + 46x^3 - 444x$

$(5x^2 - 7)(x^2 - 6)2x = 2x(5x^4 - 30x^2 - 7x^2 + 42)$ $= 2x(5x^4 - 37x^2 + 42)$ $= 10x^5 - 74x^3 + 84x$

Combining these, $f'(x) = \frac{20x^5 + 46x^3 - 444x - (10x^5 - 74x^3 + 84x)}{(x^2 + 6)^2}$ $= \frac{20x^5 + 46x^3 - 444x - 10x^5 + 74x^3 - 84x}{(x^2 + 6)^2}$ $= \frac{10x^5 + 120x^3 - 528x}{(x^2 + 6)^2}$

Step 3: Evaluate the derivative at $x = 2$

$f'(2) = \frac{10(2)^5 + 120(2)^3 - 528(2)}{((2)^2 + 6)^2}$ $= \frac{10(32) + 120(8) - 528(2)}{(4 + 6)^2}$ $= \frac{320 + 960 - 1056}{100}$ $= \frac{224}{100}$ $= 2.24$

Thus, $f'(x)$ evaluated at $x = 2$ is $2.24$.

Would you like any further details or have any questions?

Here are 8 related questions that you might find interesting:

1. How does the quotient rule differ from the product rule?
2. What are the steps for simplifying complex rational functions?
3. Can you provide more examples of functions that require the quotient rule?
4. How would you find the second derivative of $f(x)$?
5. What are some common mistakes to avoid when differentiating using the quotient rule?
6. How can we verify the result using numerical differentiation?
7. How do the critical points of $f(x)$ relate to the behavior of $f'(x)$?
8. Can the given function $f(x)$ be further simplified before differentiation?

Tip: Always double-check your algebraic manipulations to avoid errors, especially when working with derivatives.