Here is the question: "Newton’s Law of Cooling. Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings. The temperature of a warm object can be found by $T = A + (T_0 – A)e^{kt}\;$, where $A\;$ is the temperature of the surroundings, $T_0\;$ is the initial temperature of the object, $t\;$ is the time in minutes since the initial temperature was measured, and $k\;$ is a characteristic of the object. At what rate is a cup of tea cooling 3 minutes after its initial temperature was taken, if it had an initial temperature of 92°C, is placed in a room at a temperature of 21°C, and has a $k\;$ value of 0.054?"
During the explanation of the answer to this question they show this deriving:
From the four lines of math seen above, how did "they" get from the third line to the fourth line (last line)?
I believe the chain rule is being applied here, please explain the process of this math.
Thanks!
It's easier to see if you remove the units
$T(t)=21+71e^{0.054t}$
Take the derivative and you get
$T'(t)=71\cdot 0.054e^{0.054t}$
where the last step comes from the chain rule.
$(e^{at})'=e^{at}(at)'=ae^{at}$
Then it is solved.