The partial derivative

Question

Suppose the variables $x$ and $u$ are related by

$$x=u$$

Then I have a function $f=f(x)$ which does not explicitly depend on $u$.

Then is it true that $$\frac{\partial f}{\partial u}=0$$?

Answer 1

$\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial u} = f'(x)$, since $\frac{\partial x}{\partial u} = 1$.

Answer 2

No, since you have only one variable, the symbol you should be using is

$$ \frac{{\rm d}}{{\rm d}u} $$

and

$$ \require{cancel} \frac{{\rm d}f}{{\rm d}u} = \frac{{\rm d}f}{{\rm d}x}\cancelto{1}{\frac{{\rm d}x}{{\rm d}u}} = \frac{{\rm d}f}{{\rm d}x} $$