reference request – Calculation derivative of certain path integrals

Consider an F function (think of neural networks) with two sets of parameters: (1) model parameters $$mathbf {w}$$and (2) input data $${ bf x} in { mathbb R} ^ d$$. To repair $$i in [d]$$, 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, we can think of $$C_i ({ bf x}, { bf w})$$ as "contribution" of the $$i$$dimension to the final result – by integrating along a line from $$bf 0$$ at $$bf x$$.

My question is: are there any known methods (references, etc.) giving "backpropagation" algorithms for calculation:

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

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

Thank you in advance for any idea.