To solve the problem $\frac{d}{dx}(8 \cdot 3^x)$, we need to use the rule for differentiating exponential functions.

Step-by-Step Solution:

1. Recall the derivative of an exponential function: $\frac{d}{dx} \left( a^x \right) = a^x \ln(a)$ where $a$ is a constant.

2. Apply the derivative rule: Here, $a = 3$, so: $\frac{d}{dx} \left( 3^x \right) = 3^x \ln(3)$

3. Include the constant factor $8$: Since $8$ is a constant, it remains in the derivative: $\frac{d}{dx} \left( 8 \cdot 3^x \right) = 8 \cdot \frac{d}{dx} \left( 3^x \right) = 8 \cdot 3^x \ln(3)$

The correct answer is: $\textbf{(A) } 8\cdot 3^x\ln(3)$

Would you like more details on this solution, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you differentiate a function of the form $a^x$ where $a$ is a constant?
2. What is the general rule for differentiating logarithmic functions?
3. How does the derivative of a product involving a constant work?
4. Can you explain why the natural logarithm $\ln(a)$ appears in the derivative of $a^x$?
5. What happens when you differentiate an exponential function with a variable base, like $x^x$?

Tip: Remember that the natural logarithm $\ln(a)$ plays a crucial role when differentiating exponential functions with a base other than $e$.