DVD Sales Analysis Using Derivatives

In this problem, we are looking at a sales analysis function for DVDs. The total sales in thousands are given by uppercase S as a function of time t in months.

For part A, we need to find the derivative S prime of t. Since we have a fraction, we should use the quotient rule.

Let's identify our numerator u and denominator v. Here, u is ninety t squared and v is t squared plus fifty.

Now we find their derivatives. u prime is one hundred eighty t, and v prime is simply two t.

Plugging these into the quotient rule formula, we get S prime of t equals the bottom times derivative of the top minus the top times derivative of the bottom, all over the bottom squared.

Let's simplify the numerator. We distribute one hundred eighty t to get one hundred eighty t cubed plus nine thousand t. Then we subtract one hundred eighty t cubed from the second term.

The t cubed terms cancel out, leaving us with our final derivative expression.

Moving to part B, we need to evaluate the function and its derivative at t equals ten months.

Squaring ten gives one hundred. So we have nine thousand divided by one hundred plus fifty.