I am completely lost on this one, I have no idea what to do at all on it. I know that the derivative of $e^x$ is $e^x$ but that doesn't seem to help here. I am suppose to find the derivative of $y=xe^{-kx}$ I figured I could use the product rule which would be the derivative of e $e^{-kx} (x) + e^{-kx} (1)$ I need to find the derivative of $e^{-kx}$ and for that I got $-e^{-kx}$ I likely made many mistakes because my answer isn't even close.
derivatives using chain rule
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0The derivative of $e^{-kx}$ is not $-e^{-kx}$. You seemed to have pulled the *sign* out the exponent correctly, but you forget the scaling factor.. – 2011-09-22
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0I don't know what the scaling factor is, what I did though was make $y=e^u$ and $u=-kx$ is that wrong? It is problaby time I give up for the day, I can't do a single problem on my own. – 2011-09-22
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0What is the derivative of -$kx$ w.r.t $x$? – 2011-09-22
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0Jordan: I was talking about $k$ is all. Your method is completely right, but you didn't do the math correctly: what's $du/dx$? – 2011-09-22
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0@Jordan: I think that you should know that a [meta](http://meta.math.stackexchange.com/questions/4404/on-downvoting-jordan/4441#comment17072_4441) question has been brought up concerning your questions. – 2012-06-13
1 Answers
This is a combination of product rule and chain rule. To review, the product rule states
$$\frac{d f g}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}$$
or in Newton's notation
$$(fg)' = f' g + g' f.$$
And as you've identified correctly,
$$y' = (x e^{- k x})' = (1) e^{-kx} + x (e^{-kx})',$$
so we just need to determine what $(e^{-kx})'$ is. From previous questions, you know that the chain rule states that
$$\frac{df(g(x))}{dx} = \frac{df(g)}{dg} \frac{dg(x)}{dx},$$
or more succinctly
$$f'(g(x)) = f'(g) g'(x).$$
So, in $(e^{-kx})'$, we can see that $f(g) = e^g$ and $g(x) = -k x$. You can either memorize how to treat the derivative of a constant times your variable, or you could apply the product rule, again. The product rule gives
$$g'(x) = (-k)' x + -k (x)'.$$
So, what are the derivatives of $-k$ and $x$ with respect to $x$? Once you have those values in hand, plug back in.