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We are given the daily cost function \( C(x) = 10000 + 5x \), where \( x \) represents the number of switches produced daily, and we need to determine the rate of change of cost with respect to \( x \).

### Step 1: Differentiate \( C(x) \) with respect to \( x \)
The rate of change of cost with respect to \( x \) is the derivative of \( C(x) \). Let's differentiate:

\[
C'(x) = \frac{d}{dx}(10000 + 5x)
\]

Since the derivative of a constant is zero and the derivative of \( 5x \) is simply 5, we have:

\[
C'(x) = 5
\]

### Step 2: Interpretation
The rate of change of cost with respect to \( x \) is 5. This means that for every additional switch produced, the cost increases by 5 dollars.

### Step 3: Units of \( C'(x) \)
The units of \( C'(x) \) are derived from the fact that \( C(x) \) is in dollars and \( x \) represents the number of switches. Thus, the units of the rate of change are dollars per switch. This matches the option "dollars per switch".

### Final Answer:
- **Rate of change:** \( 5 \) (exact answer)
- **Units of \( C'(x) \):** Dollars per switch.

Would you like more details, or do you have any other questions?

### Related Questions:
1. What is the derivative of a general linear function \( f(x) = ax + b \)?
2. How does the rate of change differ for nonlinear cost functions?
3. If \( C(x) = 10000 + 5x^2 \), what would the rate of change of cost with respect to \( x \) be?
4. How does production scaling (increasing \( x \)) impact the total cost in general?
5. Can we model this cost function in real-time scenarios, such as inflation or material cost fluctuations?

### Tip:
For cost functions, understanding the derivative helps in predicting marginal costs—how much extra you pay for producing one more unit.

A manufacturer of precision digital switches has a daily cost C (in dollars) of C(x) = 10000 + 5x, where x is the number of switches produced daily. What is the rate of change of cost with respect to x? (Use decimal notation. Give an exact answer.)

Learn how to calculate the rate of change for the cost function C(x) = 10000 + 5x, which describes the daily cost of producing precision digital switches. This problem involves basic differentiation and is suitable for high school students. Detailed step-by-step solution included.

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