Question about differential notation ($\partial$ and $d$)

Question

My question is if, given $f(x)$, it's correct to write: $$\frac{\partial f(x)}{\partial x}$$ instead of $$\frac{d f(x)}{d x}.$$ Is it incorrect to use $\partial$ in one-variable expressions? Or it just doesn't really matter.

Answer 1

In general, we have the relation: $\frac{\mathrm d f(y)}{\mathrm d x} = \frac{\partial f(y)}{\partial y} \frac{\mathrm d y}{\mathrm d x}$.

For your setting, $y=x$, we get $\frac{\mathrm d x}{\mathrm d x} = 1$ which shows that both terms are the same: $\frac{\mathrm d f(x)}{\mathrm d x} = \frac{\partial f(x)}{\partial x}$.

(Depending on the situation, you might want to use $\frac{\mathrm d f(x)}{\mathrm d x}$ or $\frac{\partial f(x)}{\partial x}$ to emphasize what this expression represents. If you just want the derivative of the function $f$, then $\frac{\partial f(x)}{\partial x}$ is probably more cannonical.)