The given image depicts the difference quotient expression used in calculus, which is a core step in deriving the derivative of a function using limits. Let's break it down:

$\lim_{\Delta x \to 0} \frac{(5 + \Delta x)^2 - 3(2 + \Delta x) + 9 - (5^2 - 3(2) + 9)}{\Delta x}$

To clarify, this limit formula is designed to compute the derivative of a given function $f(x)$ at a specific point by using the difference quotient.

Here's a step-by-step breakdown of how to simplify and solve this expression:

Step 1: Expand the terms inside the numerator

1. Expand $(5 + \Delta x)^2$ using the binomial formula: $(5 + \Delta x)^2 = 25 + 10\Delta x + (\Delta x)^2$

2. Distribute $-3$ over $(2 + \Delta x)$: $-3(2 + \Delta x) = -6 - 3\Delta x$

Step 2: Substitute expanded forms back into the difference quotient

$\frac{(25 + 10\Delta x + (\Delta x)^2) - (6 + 3\Delta x) + 9 - (25 - 6 + 9)}{\Delta x}$

Step 3: Simplify the constant terms

The expression $25 - 6 + 9$ results in: $25 - 6 + 9 = 28$

Thus, the numerator becomes: $(25 + 10\Delta x + (\Delta x)^2) - (6 + 3\Delta x) + 9 - 28$

Step 4: Combine like terms

Simplifying the constants and combining terms involving $\Delta x$: $(25 - 6 + 9 - 28) + (10\Delta x - 3\Delta x) + (\Delta x)^2 = 0 + 7\Delta x + (\Delta x)^2$

So, the expression becomes: $\frac{7\Delta x + (\Delta x)^2}{\Delta x}$

Step 5: Simplify the expression by canceling $\Delta x$

Factor $\Delta x$ from the numerator: $\frac{\Delta x(7 + \Delta x)}{\Delta x}$

Cancel out the $\Delta x$: $7 + \Delta x$

Step 6: Take the limit as $\Delta x \to 0$

As $\Delta x \to 0$, the expression becomes: $7$

Thus, the derivative (or the value of the limit) is $7$.

Final Answer:

$\lim_{\Delta x \to 0} \frac{(5 + \Delta x)^2 - 3(2 + \Delta x) + 9 - (25 - 6 + 9)}{\Delta x} = 7$

Would you like more details on any specific steps or have any other questions?

Here are some related questions for further practice:

1. How do you find the derivative of a quadratic function in general?
2. Can the difference quotient method be applied to any function?
3. How would you apply this method to the function $f(x) = 3x^2 - 5x + 2$?
4. What does the derivative represent geometrically?
5. How do you interpret the limit in terms of instantaneous rate of change?

Tip: Always simplify and group like terms before applying the limit to make the calculations smoother.