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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.

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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

2 Answers 2

up vote 1 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!

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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?

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