https://file.aiquickdraw.com/mathbot/file/1727224505378o9t3kxjb.jpg

The sales function provided in the image is:

\[
S(t) = \frac{180t + 1}{t^2 + 6}
\]

To find the rate of change of sales when \( t = 0 \), we need to compute the derivative of \( S(t) \) with respect to \( t \) and evaluate it at \( t = 0 \).

The function is a quotient, so we will use the **quotient rule** for differentiation, which is:

\[
\frac{d}{dt} \left( \frac{f(t)}{g(t)} \right) = \frac{f'(t)g(t) - f(t)g'(t)}{(g(t))^2}
\]

Here, \( f(t) = 180t + 1 \) and \( g(t) = t^2 + 6 \).

### Steps:
1. Differentiate \( f(t) = 180t + 1 \) to get \( f'(t) \).
2. Differentiate \( g(t) = t^2 + 6 \) to get \( g'(t) \).
3. Apply the quotient rule to find \( S'(t) \).
4. Evaluate \( S'(t) \) at \( t = 0 \).

I'll now calculate this.The rate of change of sales in millions of dollars per year when \( t = 0 \) is \( 30 \) million dollars per year.

Would you like more details on the steps involved or have any other questions?

Here are 5 related questions for further practice:
1. What is the rate of change of sales when \( t = 1 \)?
2. How do we interpret the rate of change of a sales function?
3. What is the second derivative of the sales function, and what does it represent?
4. How would the rate of change of sales behave as \( t \to \infty \)?
5. Can we use any other rule besides the quotient rule for this function?

**Tip**: The quotient rule is a key tool for differentiating ratios of two functions—make sure to carefully apply it step by step!

The sales of a math performance enhancing drug are given by S(t) = (180t + 1) / (t^2 + 6), where t is in years and S(t) is in millions of dollars. What is the rate of change of sales in millions of dollars per year when t = 0?

Learn how to calculate the rate of change of sales for the function S(t) = (180t + 1) / (t^2 + 6) using the quotient rule in calculus. This problem involves finding the derivative and evaluating it at t = 0 to determine the rate of change. Suitable for advanced high school or college-level calculus students.

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