The given function is:

$f(x) = (4x + 3)^3$

Derivative of $f(x)$:

To find the derivative $f'(x)$, we'll apply the chain rule.

$f'(x) = 3(4x + 3)^2 \cdot 4$

Simplify:

$f'(x) = 12(4x + 3)^2$

Expanded Form of $f(x)$:

Expanding $(4x + 3)^3$ using the binomial theorem:

$(4x + 3)^3 = 64x^3 + 144x^2 + 108x + 27$

Would you like me to explain the expansion in more detail or solve anything else related to this function?

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Related Questions:

1. What is the second derivative $f''(x)$ of this function?
2. How would you find the critical points of $f(x)$?
3. What is the value of $f'(x)$ at $x = 1$?
4. How does the graph of $f(x)$ behave as $x \to \infty$ and $x \to -\infty$?
5. How would you find the inflection points of $f(x)$?

Tip:
The chain rule is essential when differentiating composite functions—always differentiate the outer function first, then multiply by the derivative of the inner function.