To find the derivative of \( 300(5)^{3x-1} \) with respect to \( x \), we will use the chain rule along with the derivative of an exponential function.

The derivative of \( a^u \) (where \( a \) is a constant and \( u \) is a function of \( x \)) is:
\[
\frac{d}{dx}[a^u] = a^u \ln(a) \cdot \frac{du}{dx}
\]

### Step-by-step solution:

Given function: \( f(x) = 300 \cdot (5)^{3x - 1} \)

1. **Identify the constants and exponent**:
   - The constant multiplier is \( 300 \).
   - The base of the exponential function is \( 5 \).
   - The exponent \( u(x) = 3x - 1 \).

2. **Differentiate the exponent**:
   \[
   \frac{du}{dx} = \frac{d}{dx}(3x - 1) = 3
   \]

3. **Apply the chain rule**:
   \[
   f'(x) = 300 \cdot (5)^{3x - 1} \cdot \ln(5) \cdot 3
   \]

4. **Simplify the expression**:
   \[
   f'(x) = 900 \cdot (5)^{3x - 1} \cdot \ln(5)
   \]

Thus, the derivative of \( 300(5)^{3x - 1} \) is:
\[
f'(x) = 900 \cdot (5)^{3x - 1} \cdot \ln(5)
\]

Would you like further details or clarifications on any part of the solution?

Here are 5 related questions:
1. How do you find the derivative of a general exponential function?
2. What is the chain rule and when do we use it?
3. How do logarithmic functions play a role in differentiating exponential functions?
4. How do we find the second derivative of \( 300(5)^{3x-1} \)?
5. What are some practical applications of derivatives involving exponential growth?

**Tip**: When differentiating exponential functions, always remember to multiply by the natural logarithm of the base.

Discover how to find the derivative of 300(5)^(3x-1) using the chain rule and the properties of exponential functions. This step-by-step solution breaks down the differentiation process and applies logarithmic functions, suitable for Grades 11-12 students.

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