Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

I have a very simple factoring question; I'm doing a calculus problem in which part of the question requires me to factor a derivative. The derivative in question is $e^{-x}tx^{t-1}-e^{-x}x^t$ (the derivative of $\frac{x^t}{e^x})$. I have no problem with finding the derivative, and once the derivative is factored I can easily solve the problem, but I embarrassingly can't figure out how to factor the derivative by hand into the form $-e^{-x}x^{t-1}(x-t)$. I suspect my problem is that I'm running on rote muscle memory of factoring polynomials. I would appreciate a quick walk-through of the hand computations.

share|cite|improve this question
The formatting is a mess -what is $x(t)$ for example? – Thomas Andrews Apr 23 '13 at 17:15
Presumably, you mean $-e^{-x}(x^{t-1})(x-t)$ in the third formula. – Thomas Andrews Apr 23 '13 at 17:17
up vote 2 down vote accepted

Starting with $e^{-x}tx^{t-1}-e^{-x}x^t$

you probably definitely proceeded to $e^{-x}(tx^{t-1}-x^t)$

and then maybe you are overlooking that $x^t=x\cdot x^{t-1}$ so that you recognize both terms hold a factor of $x^{t-1}$.

I've seen this kind of blindness before when students struggle to factor things like $x^{1/2}+x^{3/2}$. They sometimes don't immediately see that $x^{1/2}$ is a common factor since $x^{3/2}=x\cdot x^{1/2}$. It's a good thing to be aware of!

share|cite|improve this answer
Thank you, this was exactly what I was overlooking. Thanks also to @CameronWilliams for correcting the formatting. – DeusExCinema Apr 24 '13 at 1:16

I prefer using product rule and so I'd rewrite it as $x^te^{-x}$. The derivative of this is $tx^{t-1}e^{-x}-x^te^{-x}$. Both terms have a common factor of $x^{t-1}e^{-x}$ so we can factor that out to get $x^{t-1}e^{-x}(t-x)$. This is as far as it can be factored without further complicating the expression. What seems to be giving you trouble later in the problem?

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.