# Back-propagation for computing derivative of certain line integral

Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $$\mathbf{w}$$, and (2) input data $${\bf x} \in {\mathbb R}^d$$. Fix $$i \in [d]$$, consider the following path integral:

$$C_i({\bf x}, {\bf w}) = {\bf x}_i \int_0^1\frac{\partial F}{\partial {\bf x}_i}(\alpha {\bf x})d\alpha$$

Basically, one can think of $$C_i({\bf x}, {\bf w})$$ as the "contribution" of the $$i$$-th dimension to the final output -- by integrating along a line from $$\bf 0$$ to $$\bf x$$.

My question is whether there are known methods (references, etc.) that give "backpropagation-like" algorithms for computing:

$$\frac{\partial C_i({\bf x}, {\bf w})}{\partial {\bf w}_i}$$

That is, basically I want to understand the first order information of the contribution of the $$i$$-th dimension w.r.t. $${\bf w}_i$$.

Thanks in advance for any insight.