Problem

**Find \[\dfrac{d}{dx}(8\cdot3^x)\].**

Choose 1 answer:

Choose 1 answer:

(Choice A)

\[8\cdot 3^x\ln(3)\]

A

\[8\cdot 3^x\ln(3)\]

(Choice B)

\[8\cdot 3^{x-1}\]

B

\[8\cdot 3^{x-1}\]

(Choice C)

\[8\cdot 3^x\ln(x)\]

C

\[8\cdot 3^x\ln(x)\]

(Choice D)

\[8\cdot 3^x\log_3(x)\]

D

\[8\cdot 3^x\log_3(x)\]

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 \(e\).

Learn how to find the derivative of 8 * 3^x with a detailed step-by-step solution. This problem involves differentiating exponential functions and applying the derivative rule. Suitable for advanced high school students and anyone studying calculus.

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