  The estimation of causal effects is a fundamental goal in the field of causal
inference. However, it is challenging for various reasons. One reason is that
the exposure (or treatment) is naturally continuous in many real-world
scenarios. When dealing with continuous exposure, dichotomizing the exposure
variable based on a pre-defined threshold may result in a biased understanding
of causal relationships. In this paper, we propose a novel causal inference
framework that can measure the causal effect of continuous exposure. We define
the expectation of a derivative of potential outcomes at a specific exposure
level as the average causal derivative effect. Additionally, we propose a
matching method for this estimator and propose a permutation approach to test
the hypothesis of no local causal effect. We also investigate the asymptotic
properties of the proposed estimator and examine its performance through
simulation studies. Finally, we apply this causal framework in a real data
example of Chronic Obstructive Pulmonary Disease (COPD) patients.
