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I seem to have completely lost my bearing with implicit differentiation. Just a quick question:

Given $y = y(x)$ what is $$\frac{d}{dx} (e^x(x^2 + y^2))$$

I think its the $\frac d{dx}$ confusing me, I don't what effect it has compared to $\frac{dy}{dx}$. Any help will be greatly appreciated.

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up vote 2 down vote accepted

$d/dx$ means "derivative with respect to x", with implicit differentiation, you assume $y$ is a function of $x$, i.e. $y=y(x)$, so that , $y'$ is meant to mean $dy/dx$.

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It is $$e^x(x^2 + y^2) + e^x(2x + 2yy'),$$ where I use $y'$ for $dy/dx$.

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