Given a $d$-dimensional Gaussian $X \sim N(\mu, \Sigma)$ and two real-valued differentiable functions $f,g$ with bounded first derivatives, I am wondering if there is a simple expression for the covariance:
$$ \text{Cov} (f(X),g(X)). $$ For example, when $f(X) = X_k$ for some $k \in \{1,\dots, d\}$, Stein's lemma gives
$$ \text{Cov} (X_1,g(X)) = \sum_{i=1} \Sigma_{1i} \mathbb{E} \left( \frac{\partial g(x)}{\partial x_i} \bigg |_{x=X}\right). $$
Are there similar expressions for more general $f$ though?
reference: Lemma 1 here https://reader.elsevier.com/reader/sd/pii/016771529490121X?token=62B2E8B7BA323BC1AF5B699E9A3115C156EC43A959F764C79D0C044730A2C8F605B5CC1F9E5807764BCC2D37793AC619&originRegion=us-east-1&originCreation=20220630190410