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The question visible in the image is about finding the **instantaneous rate of change** for the function \( f(x) = 6x^2 - 4 \) at different values of \( x \).

### Instantaneous Rate of Change:
This can be calculated by taking the **derivative** of the function \( f(x) \), as the derivative represents the slope of the tangent line at any given point, i.e., the rate of change.

1. **Differentiate \( f(x) \):**

   The function given is:
   \[
   f(x) = 6x^2 - 4
   \]

   To find the derivative \( f'(x) \):
   \[
   f'(x) = \frac{d}{dx}(6x^2) - \frac{d}{dx}(4)
   \]
   Using the power rule \( \frac{d}{dx}(x^n) = n x^{n-1} \), we get:
   \[
   f'(x) = 12x
   \]

2. **Evaluate the derivative at the given points:**

   - For \( x = -2 \):
     \[
     f'(-2) = 12(-2) = -24
     \]
   
   - For \( x = 0 \):
     \[
     f'(0) = 12(0) = 0
     \]
   
   - For \( x = 4 \):
     \[
     f'(4) = 12(4) = 48
     \]
   
   - For \( x = 8 \):
     \[
     f'(8) = 12(8) = 96
     \]

### Summary of Results:
- Instantaneous rate of change at \( x = -2 \): **-24**
- Instantaneous rate of change at \( x = 0 \): **0**
- Instantaneous rate of change at \( x = 4 \): **48**
- Instantaneous rate of change at \( x = 8 \): **96**

Would you like further details or explanations?

Here are 5 related questions you might find useful:
1. What is the significance of the derivative in finding the instantaneous rate of change?
2. How do you compute the derivative of higher-order polynomial functions?
3. Why is the instantaneous rate of change zero at \( x = 0 \)?
4. How does the value of \( x \) affect the steepness of the curve?
5. How can this concept be applied to real-world problems, like velocity or population growth?

**Tip:** Remember that the derivative gives the slope of the tangent line, which tells you how the function is changing at any specific point.

For the function f(x) = 6x^2 - 4, estimate the instantaneous rate of change for the given values of x: a) x = -2, b) x = 0, c) x = 4, d) x = 8.

Learn how to estimate the instantaneous rate of change for the quadratic function f(x) = 6x^2 - 4 at specific points like x = -2, x = 0, x = 4, and x = 8. This example uses differentiation and the power rule, making it ideal for students in Grades 10-12.

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