Notation of derivative w.r.t. the multi-variable function argument of an univariate function
Solution 1:
Should I write the derivative of $f$ as $\frac{df}{dg}$ or $\frac{\partial f}{\partial g}$?
Both are correct. When the function is monovariate, its partial derivative is the total derivative, (with respect to its argument). The former notation is usually preferred because is clearly indicates the derivative has a few properties which a multivariate partial derivative lacks.