Is this operation a partial or total derivative? I have stacked some functions as follows:
$$ \lambda(g(J))$$
and now I have computed the derivative as 
$$\lambda'(g(J))g'(J)$$
I would like to use a short hand for this. I'm torn between 
$$ \frac{\partial \lambda(J)}{\partial J}\\
\frac{d \lambda(J)}{dJ}$$
It seems to me that a partial derivative in the current form should wrong, as $J$ does not directly appear in $\lambda(g)$. On the other hand, I remember having seen first order conditions (where we take the partial derivative) that arrive at the first derivative as in my second equation.
 A: I think either is fine. Personally, I think it's better to use $\partial$ as often as possible, since $d$ can get a little overloaded with other types of derivatives.
For the partial derivative, we usually consider it to "hold the other variables constant and ignore them". If there's just one variable, well, the "other" variables are indeed not really variable and can be ignored by virtue of not existing. Using the standard chain rule:
$$
\frac{\partial\lambda}{\partial J}
=
\frac{\partial\lambda}{\partial g}\frac{\partial}{\partial J}g(J)
=
\frac{\partial\lambda}{\partial g}\frac{\partial g}{\partial J}
\frac{\partial J}{\partial J}
=
\frac{\partial\lambda}{\partial g}\frac{\partial g}{\partial J}
$$
The use of $\partial$ in $\frac{\partial\lambda}{\partial g}$ makes sense especially here since we are treating $g$ as an independent variable rather than a function of $J$ (i.e. ignoring $J$).
On the other hand, I think most people think of the total derivative as not being with respect to a particular variable, so:
$$
d\lambda = \frac{\partial \lambda}{\partial g}\frac{\partial g}{\partial J}dJ
$$
so you can notationally divide by the differential to get:
$$
\frac{d\lambda}{dJ}= \frac{\partial \lambda}{\partial g}\frac{\partial g}{\partial J}
$$
I also like the notation $\partial_J\lambda=\partial_g\lambda\partial_Jg$ to avoid the fractions.
