Let $f : \mathbb{C} \rightarrow \mathbb{C}$ and $\gamma : [a,b]\subset \mathbb{R} \rightarrow \mathbb{C}$ be differentiable functions.
Can i apply the chain rule on the derivative of $f(\gamma (t))$ with respect to $t$?
i.e. $\dfrac{d}{dt} f(\gamma (t)) = f'(\gamma (t)).\gamma'(t)$
Why is this correct, or why is this wrong?
If this is wrong, then should i apply the Cauchy-Riemann equation instead?
Yes, it is correct. And quite useful.
On the other hand, no, you cannot apply the Cauchy-Riemann equations instead. Actually, this makes no sense, because the Cauchy-Riemann equations are for complex functions defined on an open subset of $\mathbb C$, whereas the domain of $f\circ\gamma$ is $[a,b]$.