I'm trying to prove the following generalization of Leibniz's Rule:
Let $\Omega$ be an open set and let $\gamma$ be a rectifiable curve in $\mathbb{C}$. Suppose that $\varphi:\{\gamma\}\times\Omega\to\mathbb{C}$ (where $\{\gamma\}$ is the trace of $\gamma$) is a continuous function and define $g:\Omega\to\mathbb{C}$ by $$g(z)=\int_{\gamma}\varphi(w,z)\:dw.$$ Then $g(z)$ is continuous. If $\frac{\partial\varphi}{\partial z}$ exists for each $(w,z)$ in $\{\gamma\}\times\Omega$ and is continuous, then $g(z)$ is analytic and $$g'(z)=\int_{\gamma}\frac{\partial\varphi}{\partial z}(w,z)\:dw.$$
A similar question has already been asked here:
However, as far as I can tell, it is a proof of the Leibniz Rule itself, not the desired analogue. I think the answer in the question above is very helpful. Would the approach taken in the answer be valid when considering the domain $\{\gamma\}\times\Omega$ of $\varphi$ rather than the domain $[a,b]\times D$ of $f$ in the above question? Thanks for any help!