Using Leibniz Notion Chain Rule: Solve the rate of change of power

Question

The amount of power, P in Watts, needed to run a particular machine is given by P(I) = 20I^2 where I is the current in Amps, which changes over time with the formula: I(t) = 4t +5 / t + 3 where t is in minutes. Find the rate of change of the power when the machine is running for 4 minutes. 

(I think we're going to use Leibniz Notation Chain Rule)

Here's what I did but I don't know if I did it right:

We are given the formulas: P(I) = 20I^2 and I(t) = 4t + 5/t + 3

Since the amount of power, P in Watts, needed to run a particular machine is given by P(I) = 20I^2 where I is the current in Amps, and the formula which determines the changes over time is I(t) = 4t + 5/t + 3, where t is in minutes. We can substitute in both formulas together and then substitute in t = 4. Therefore, it would look like this:

P(I) = 20(4t + 5/t + 3)^2

Substitute in 4 minutes for t:

P(4t + 5/t + 3) = 20(4(4) + 5/(4) + 3)^2

P(4t + 5/t + 3) = 8201.25

Therefore, the rate of change of the power when the machine is running for 4 minutes is 8201.25

Answer 1

If I understand the problem correctly, it is asking for

dP/dt, the rate of change in power per minute

P(t) = 20I²(t)

dP/dt = 40I(t)dI(t)/dt Using the chain rule

dP/dt = 40(4t-5/t+3)(4-5/t²)

dP(4)/dt = 40(16-5/4+3)(4-5/16) = 2618.125 Watts/minute