I want to differentiate this line integral with respect to one of the endpoints. The integral is computed over the segment between $\mathbf{x}_i$ and $\mathbf{x}_{i+1}$.
\begin{equation} \frac{\partial}{\partial \mathbf{x}_i}\int_{\mathbf{x_i}}^{\mathbf{x}_{i+1}} \mathbf{A}\cdot d\mathbf{l}, \end{equation}
where $\mathbf{A}$ is a 'smooth' vector field with no irregularity or differentiability problem on the segment $\{\mathbf{x}(t):=(\mathbf{x}_{i+1}-\mathbf{x}_i)t + \mathbf{x}_i,t\in[0,1]\}$.
Wanted to use at first an analogous expression to Leibniz integral rule, but was confused on how to use it here, so I decided to perturbate one of the endpoints by a term $\epsilon \delta \mathbf{x}$, and take the limit as $\epsilon$ goes to zero of the standard quotient as below:
\begin{equation} \begin{split} &\frac{\partial}{\partial \mathbf{x}_i} \int_{\mathbf{x}_i}^{\mathbf{x}_{i+1}} \mathbf{A}(\mathbf{x})\cdot d\mathbf{l} \\ =&\lim_{\epsilon \downarrow 0}\frac{1}{\epsilon}\Bigg\{ \int_{0}^{1} \mathbf{A}(t(\mathbf{x}_{i+1} - \mathbf{x}_i - \epsilon \delta \mathbf{x}_i) + \mathbf{x}_i + \epsilon \delta \mathbf{x}_i) \cdot (\mathbf{x}_{i+1} - \mathbf{x}_{i} - \epsilon \delta \mathbf{x}_i) dt\\ &-\int_{0}^{1} \mathbf{A}(t(\mathbf{x}_{i+1} - \mathbf{x}_i) \cdot (\mathbf{x}_{i+1} - \mathbf{x}_{i}) dt \Bigg\}\\ \end{split} \end{equation}
Then I obtain a result which depends on my perturbation $\delta \mathbf{x}_i$ through a directional derivative of $\mathbf{A}$. What do yo think? I'd like to have an expression that doesn't, if possible, depends on that perturbed quantity.
\begin{equation} \frac{\partial}{\partial \mathbf{x}_i} \int_{\mathbf{x}_i}^{\mathbf{x}_{i+1}} \mathbf{A}(\mathbf{x})\cdot d\mathbf{l} = \int_{\mathbf{x}_i}^{\mathbf{x}_{i+1}} J_{\mathbf{A}}(\mathbf{x})\delta \mathbf{x}_i \cdot d\mathbf{l} - \int_{0}^{1} \mathbf{A}(\mathbf{x}(t))\cdot \delta \mathbf{x}_idt. \end{equation}
Note however the similarity with the Leibniz integration rule in the form of the result.
Thank you in advance for your comments. Best