Gauge invariance for the electric field

We know that $$boldsymbol{E}=-nabla V-frac{partialboldsymbol{A}}{partial t}$$

$$boldsymbol{B}=nablatimesboldsymbol{A}$$
But I see that the following changes do not change these fields:
$$boldsymbol{A}toboldsymbol{A}+nabla f$$
$$Vto V-frac{partial f}{partial t}$$

My question is about $V$, why is the partial derivative with respect to time does not change the electric field? why would $$nablaleft(frac{partial f}{partial t}right)=0$$
I don’t see that explained anywhere, and I fail to understand how that is so trivial for any function $f$ that depends on position and time.