we found that the derivative of the function is given by 

*f* *'*(*x*) = 1/4.  The domain of the derivative is as follows.

Given that the derivative of the function is \( f'(x) = \frac{1}{4} \), we can interpret a few things about this function and its domain.

1. **Constant Derivative:** Since the derivative \( f'(x) = \frac{1}{4} \) is a constant, the original function \( f(x) \) must be linear. This is because the derivative of a linear function \( f(x) = \frac{1}{4}x + C \) is constant.

2. **Domain of the Derivative:** Since \( f'(x) \) is constant and defined as \( \frac{1}{4} \), it has no restrictions on its domain. This means the derivative exists for all real values of \( x \), implying the domain of the derivative is:
   \[
   \text{Domain of } f'(x): \mathbb{R} \quad \text{(all real numbers)}
   \]

In conclusion, the domain of the derivative \( f'(x) = \frac{1}{4} \) is all real numbers, \( (-\infty, \infty) \).

Would you like more details or have any other questions?

Here are 5 related questions to explore further:
1. What is the original function \( f(x) \) if \( f'(x) = \frac{1}{4} \)?
2. How can you find the slope of a linear function from its derivative?
3. What is the geometric interpretation of a constant derivative?
4. How does the domain of the derivative compare to the domain of the original function?
5. Can a derivative be constant for non-linear functions?

**Tip:** A constant derivative always indicates a linear function, where the slope is uniform across all values of \( x \).

We found that the derivative of the function is given by f'(x) = 1/4. The domain of the derivative is as follows.

Explore the derivative f'(x) = 1/4, how it indicates a linear function, and why its domain is all real numbers (-∞, ∞). This detailed explanation includes key concepts in calculus and linear functions, ideal for Grades 11-12 or early college students.

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