We have the function from the image:

$y = \frac{\sqrt{x^2 + 7}}{x - 3}$

To find the derivative of this function using the quotient rule (often referred to as the "10-step rule"), we break it down as follows.

Step-by-Step Process:

Step 1: Quotient Rule Formula

The derivative of a quotient $\frac{f(x)}{g(x)}$ is given by:

$y' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

Here, $f(x) = \sqrt{x^2 + 7}$ and $g(x) = x - 3$.

Step 2: Derivative of $f(x)$

The function $f(x) = \sqrt{x^2 + 7}$ is a composite function. To differentiate it, we apply the chain rule.

$f'(x) = \frac{1}{2\sqrt{x^2 + 7}} \cdot 2x = \frac{x}{\sqrt{x^2 + 7}}$

Step 3: Derivative of $g(x)$

The derivative of $g(x) = x - 3$ is straightforward:

$g'(x) = 1$

Step 4: Apply the Quotient Rule

Now, we apply the quotient rule using the derivatives from steps 2 and 3.

$y' = \frac{\left( \frac{x}{\sqrt{x^2 + 7}} \right)(x - 3) - \left( \sqrt{x^2 + 7} \right)(1)}{(x - 3)^2}$

Step 5: Simplify the numerator

First, expand the terms in the numerator:

$y' = \frac{x(x - 3)}{\sqrt{x^2 + 7}} - \sqrt{x^2 + 7}$

This simplifies to:

$y' = \frac{x^2 - 3x}{\sqrt{x^2 + 7}} - \sqrt{x^2 + 7}$

Step 6: Combine the terms over a common denominator

Now we combine the terms under a common denominator:

$y' = \frac{x^2 - 3x - (x^2 + 7)}{\sqrt{x^2 + 7}(x - 3)^2}$

Simplifying the numerator:

$y' = \frac{x^2 - 3x - x^2 - 7}{\sqrt{x^2 + 7}(x - 3)^2}$

$y' = \frac{-3x - 7}{\sqrt{x^2 + 7}(x - 3)^2}$

Final Answer:

The derivative of the function $y = \frac{\sqrt{x^2 + 7}}{x - 3}$ is:

$y' = \frac{-3x - 7}{\sqrt{x^2 + 7}(x - 3)^2}$

Would you like further clarification or additional steps explained?

Here are 5 related questions that might interest you:

1. How do we apply the chain rule in complex functions?
2. What are common mistakes to avoid when using the quotient rule?
3. How can we interpret the behavior of the derivative at $x = 3$?
4. How do we handle square roots in differentiation?
5. How do we find the second derivative of this function?

Tip: Always simplify your functions before applying the quotient rule to avoid unnecessary complications during the derivative process.