Question 987918
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(y)\ =\ 4x^2\ +\ 8y^2\ =\ 36]


is not a function.  If you graph the ellipse, it fails the vertical line test.  So I presume what you really mean is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(y)\ =\ 4x^2\ +\ 8y^2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dy}4x^2\ +\ 8y^2\ =\ 16y]


Because when considering the expression as a function of *[tex \Large y], any expression involving only *[tex \Large x] is considered a constant.  This becomes a simple application of the power rule.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2}{dy^2}4x^2\ +\ 8y^2\ =\ \frac{d}{dy}16y\ =\ 16]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \