Δx
Δy
​
 = 
Δx
(24x 
2
 +48xΔx+24(Δx) 
2
 +32)−(24x 
2
 +32)
​
 


The given expression represents the difference in function values over the difference in \( x \), typically used to find the derivative in calculus. Let’s break down the expression step-by-step:

We have the expression:

\[
\frac{\Delta y}{\Delta x} = \frac{(24x^2 + 48x\Delta x + 24(\Delta x)^2 + 32) - (24x^2 + 32)}{\Delta x}
\]

### Step-by-Step Breakdown:

1. **Simplify the numerator:**
   The numerator is \( (24x^2 + 48x\Delta x + 24(\Delta x)^2 + 32) - (24x^2 + 32) \).

   Subtract the second expression inside the parentheses:
   \[
   24x^2 + 48x\Delta x + 24(\Delta x)^2 + 32 - 24x^2 - 32
   \]

   After cancellation, we are left with:
   \[
   48x\Delta x + 24(\Delta x)^2
   \]

2. **Divide by \( \Delta x \):**
   Now, divide the simplified numerator by \( \Delta x \):
   \[
   \frac{48x\Delta x + 24(\Delta x)^2}{\Delta x}
   \]

   Simplify the terms by factoring out \( \Delta x \):
   \[
   48x + 24\Delta x
   \]

### Final Simplified Expression:

\[
\frac{\Delta y}{\Delta x} = 48x + 24\Delta x
\]

As \( \Delta x \) approaches 0 (in the context of finding the derivative), the second term \( 24\Delta x \) vanishes. So the derivative (or slope of the tangent line) is:

\[
\lim_{\Delta x \to 0} \left( 48x + 24\Delta x \right) = 48x
\]

This suggests that the derivative of the original function is \( 48x \).

---

Would you like further details on any specific steps or concepts?

### Related Questions:
1. What is the significance of the derivative in calculus?
2. How does the limit \( \Delta x \to 0 \) simplify expressions in derivative calculations?
3. Can you explain how the power rule applies to derivatives like this?
4. How does this derivative relate to the original function?
5. What are some applications of derivatives in physics or economics?

### Tip:
When simplifying differences of functions to find derivatives, always look for common terms that cancel out early to streamline the process.

Δy/Δx = [(24x^2 + 48xΔx + 24(Δx)^2 + 32) - (24x^2 + 32)] / Δx

Learn how to use the limit definition of a derivative to solve a polynomial expression step-by-step. This problem involves simplifying a difference quotient and finding the derivative of a quadratic function, suitable for high school calculus students.

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