What is the instantaneous rate of change given these business numbers?

Question

-A man has 61 clients

-Clients are growing at a rate of 2 per year.

-His profits are 631.42 per client per year.

-His profits are dropping by 12.41 per client per year.

 What's the instantaneous rate of change for his profits?

Answer 1

Hello, thank you for taking the time to post your question!

This one looks like a complicated solving process but it’s actually just a simple application of the Product Rule. We have the general equation that Profits = Clients x Profit/Client, so applying the product rule and then plugging in the given numbers gives

dT/dt = (N * dP/dt) + (P * dN/dt)

dT/dt = 61(-12.41) + 631.42(2)

dT/dt = 506.83

So the instantaneous rate of change for his profits is approximately $506.83 dollars per year

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)