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.