The derivative of the given function $f(x) = x^2 \cdot 7^x$ is:

$f'(x) = 7^x \cdot x \cdot (x \ln(7) + 2)$

Would you like a more detailed explanation of the steps involved in differentiating this function?

Here are 5 related questions to expand on this concept:

1. How do you differentiate products of functions (Product Rule)?
2. What is the logarithmic differentiation technique and when is it useful?
3. How do you differentiate exponential functions like $a^x$?
4. What is the Chain Rule and how does it apply here?
5. How does the derivative change if the function is $x^n \cdot a^x$, where $n$ is any power?

Tip: Always identify when to use the product and chain rules when differentiating products involving exponential and polynomial functions.