Working in Schrodinger picture, while deriving Ehrenfest's theorem, we go - $$ \frac{d}{d t}\langle A\rangle=\frac{d}{d t}\langle\psi|\hat{A}| \psi\rangle $$ $A$ is an operator. Expanding RHS- $$ \frac{d}{d t}\langle A\rangle=\left\langle\frac{d}{d t} \psi|\hat{A}| \psi\right\rangle+\left\langle\psi\left|\frac{\partial}{\partial t} \hat{A}\right| \psi\right\rangle+\left\langle\psi|\hat{A}| \frac{d}{d t} \psi\right\rangle $$ My doubt is regarding the second term. Why do we write $\frac{\partial}{\partial t}\hat{A}$ and not $\frac{d}{d t}A$? Of course, this notation wouldn't matter incase there is only an explicit dependence of $t$, if there's any $t$ dependence at all.
What if $A$ were composed of other time dependent operator $\hat{O}(t)$, i.e. $\hat{A}(t)=A(\hat{O}(t),t)$. Can we have such operators? In that case $\frac{\partial}{\partial t}\hat{A} \neq\frac{d}{d t}A$.