Notation of derivative w.r.t. the multi-variable function argument of an univariate function

Question

Suppose I have a univariate function, whose variable is a multi-variable function, i.e. $$ f=f(g)\,,\quad g=g(x,y)\,.$$

Should I write the derivative of $f$ as $\frac{df}{dg}$ or $\frac{\partial f}{\partial g}$?

When using chain rule, which nontation is correct or better?

$$\frac{\partial f}{\partial x}=\frac{d f}{d g}\frac{\partial g}{\partial x}$$

or

$$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$

I saw the second notation (all partial derivative) on a book about solid mechanics, but $\frac{df}{dg}$ makes more sense to me.

Answer 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.