https://file.aiquickdraw.com/mathbot/file/1726208896398dnu5kxtt.jpg

The expression provided in the image involves the limit:

\[
\lim_{h \to 0} \frac{(8 + h)^3 - 512}{h}
\]

This is the definition of the derivative of a function \( f(x) \) evaluated at some point \( a \). Here, \( f(x) = x^3 \) and the point \( a \) at which we evaluate the derivative is 8. We can recognize this because \( 512 = 8^3 \).

To find the derivative \( f'(x) \) of the function \( f(x) = x^3 \):

1. Differentiate \( f(x) = x^3 \) with respect to \( x \).
2. Using the power rule \( \frac{d}{dx} x^n = nx^{n-1} \), we have:
   \[
   f'(x) = 3x^2
   \]
3. Evaluate \( f'(x) \) at \( x = 8 \):
   \[
   f'(8) = 3 \times 8^2 = 192
   \]

Thus, the answers are:
- \( f(x) = x^3 \)
- \( a = 8 \)
- The value of the limit is \( 192 \).

Would you like to know more about the process of finding derivatives or have any other questions?

Further queries you might be interested in:
1. How do you calculate higher order derivatives of \( x^3 \)?
2. What happens to the derivative as \( x \) becomes very large?
3. How would the derivative change if the function were \( f(x) = x^3 + 5x \)?
4. Can you explain the geometric interpretation of the derivative at a point?
5. What are some real-world applications of calculating such derivatives?

**Tip:** When studying calculus, it can be very helpful to practice the mechanical aspects of taking derivatives alongside understanding their conceptual implications to gain both fluency and insight into their applications.

The limit lim_{h -> 0} [(8 + h)^3 - 512] / h is the standard expression defining the derivative of some function f(x) at some number a. Find f(x) and a.

Solve the limit expression lim_{h -> 0} [(8 + h)^3 - 512] / h, which defines the derivative of the cubic function f(x) = x^3 at the point a = 8. This step-by-step explanation is suitable for high school and college-level students studying calculus.

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