Derivative of Heaviside step function multiplied by an exponential

Question

Let $f(y)=e^y\mathscr{H}(y)$ where $\mathscr{H}$ is the Heaviside step function. We know that the derivative of $\mathscr{H}$ is given by $$\frac{d}{dy}\mathscr{H}(y)=\delta(y),$$ Then how come, in both a paper (Bouchouev, Isakov - The inverse problem of option pricing) and the output of Wolfram Alpha has the following $$\frac{d}{dy}f(y)=\frac{d}{dy}(e^y\mathscr{H}(y))=\delta(y)+e^y\mathscr{H}(y),$$ rather than what you'd expect from the product rule $$\frac{d}{dy}f(y)=\frac{d}{dy}(e^y\mathscr{H}(y))=e^y\delta(y)+e^y\mathscr{H}(y).$$