I am asked to apply the quotient rule $$f(x)=\frac{p(x)}{q(x)}$$ and show that $$f'(0)=\frac{p'(0)}{q(0)}$$ if $p(0)=0$, and hence evaluate $f'(0)$ where $$f(x)=\frac{xe^{2x}}{(2-x)(1-x)^2}$$ ...dont get it...just replace all $x$'s by $0$?? I don't get what they ask me to show then...:S
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Please check my edit. I hope this is correct. – Julien Feb 11 '13 at 22:30
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Note that you need the condition $q(0)\neq 0$ for this to be true. It holds for your example. You can't just replace all $x$'s by $0$, you need to compute $p'$ first. – Julien Feb 11 '13 at 22:31
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If $f(x) = p(x)/q(x)$, then $f'(x) = \frac{p'(x)q(x)-p(x)q'(x)}{q^2(x)}$.
If $p(0) = 0$, then $f'(0) = \frac{p'(0)q(0)-0\cdot q'(0)}{q^2(0)}$.
You should be able to take it from here.
Emily
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The quotient rule tells you that $$f'(x) = \frac{p'(x)}{q(x)} - \frac{p(x)q'(x)}{q(x)^2}.$$
If $p(0) = 0$ and $q(0) \neq 0$, what is $f'(0)$?
The second part is just putting this principle into practice for a concrete example.
user7530
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