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For instance, $f=x^2y^2+1$ is an element of $\mathbb{C}[x,y]$, I want to extract the coefficient of $x^2$, which is $y^2$. However "f.coefficient(x,2)" only works for symbolic expressions, are there similar functions for ring elements?

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I don't know anything about Sage, but perhaps you could tell it to find $\dfrac{1}{2}\dfrac{\partial^2 f}{\partial x^2}(0,y)$? – Jonas Meyer Jun 30 '12 at 7:51
By the way, there ask.sagemath.org for sage. In case you have more specific questions in the future. – Jernej Jun 30 '12 at 8:59

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Simply use f.coefficient(x^2), as in this example:

sage: R.<x,y> = CC[]
sage: f = x^2 *  y^2 + 1
sage: f.coefficient(x^2)
y^2
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