Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$$f(t) = \frac{100e^t}{10+e^t} \text{for all } t \in \mathbb{R}$$

(1) show that $$\frac{d}{dt}f(t)=f(t) - \frac{1}{100} (f(t)^2)$$ (2) waht is the second derivative $f^{(2)}(t)$ in term of $f(t)$?

(3) find the second order Taylor polynomial $T_2(t)$ for $f(t)$ about the point $t = \ln(10)$.

I really donot know how to solve this question.

share|cite|improve this question
Let's start at the beginning. Can you find $\frac{d}{dt} f(t)$? – Hurkyl Feb 23 '13 at 6:57

A nifty trick you might try is to observe that

$$\frac{f'(t)}{f(t)} = \frac{d}{dt} \log{f(t)}$$

You get a form like the left hand side by dividing both sides of the equation by $f(t)$. Your equation to verify is then

$$\frac{d}{dt} \log{f(t)} = 1-\frac{1}{100} f(t)$$


$$\log{f(t)} = \log{100} - \log{(1+10 e^{-t})}$$

$$\frac{d}{dt} \log{f(t)} = \frac{10 e^{-t}}{1+10 e^{-t}}$$

Can you do the rest?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.