Derivative commutes with the coercion $\mathbb{R} \hookrightarrow \mathbb{C}$: $\dfrac{d}{dy}\,\bigl(f(y) : \mathbb{C}\bigr) = f'(y)$

Let $f : \mathbb{R} \to \mathbb{R}$ be any real function and let $z \in \mathbb{R}$. Consider the complex-valued function obtained by post-composing $f$ with the embedding of $\mathbb{R}$ into $\mathbb{C}$, i.e. $y \mapsto (f(y) : \mathbb{C})$. Then its derivative at $z$ is the coercion of the real derivative:

where the right-hand side is the real derivative of $f$ at $z$, viewed as a complex number. No differentiability hypothesis is needed: the coercion $\mathbb{R} \to \mathbb{C}$ is a continuous linear isometric embedding, so it maps derivatives to derivatives, and when $f$ is not differentiable at $z$ both sides are the junk value $0$.

This is a small but ubiquitous bridging lemma for computations that mix real-variable calculus with complex-valued integrands, as happens throughout the Fourier/Mellin and contour-integration infrastructure of the PNT+ project.