What is d/dtlog P(t)?

Question 70

MediumOriginal question

A town's population $t$ years from now is modelled by $P(t) = 50{,}000\,e^{0.02t}$ .

What is $\frac{d}{dt}\log P(t)$ ?

\frac{1}{50{,}000\,e^{0.02t}}

Correct answerYour answer

1000\,e^{0.02t}

Correct answerYour answer

0.02t

Correct answerYour answer

0.02

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\log P = \log 50{,}000 + 0.02t$ .
So $\frac{d}{dt}\log P = 0.02$ at every $t$ : each year the population grows by the same fraction of itself.

Worth noting

$\frac{1}{P}$ is the derivative of $\log u$ without the factor $u' = P'(t)$ .

Ten questions from Chain Rule, Log Differentiation & Rate of Change, at your own pace.

Ten questions from Chain Rule, Log Differentiation & Rate of Change, at your own pace.