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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1$\begingroup$ The formatting is a mess -what is $x(t)$ for example? $\endgroup$ – Thomas Andrews Apr 23 '13 at 17:15
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$\begingroup$ Presumably, you mean $-e^{-x}(x^{t-1})(x-t)$ in the third formula. $\endgroup$ – Thomas Andrews Apr 23 '13 at 17:17
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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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$\begingroup$ Thank you, this was exactly what I was overlooking. Thanks also to @CameronWilliams for correcting the formatting. $\endgroup$ – DeusExCinema Apr 24 '13 at 1:16
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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?
answered Apr 23 '13 at 17:18
