Derivative of Dirac $\delta$ times a smooth function

Question

Suppose we have a test function $\phi$ and the expression: $$\int_\mathbb{R}\phi(x)\delta(x-a)dx $$

next, we would like to multiply by a $h(x)\in C^\infty$ function and then take the derivative of the combined expression. which of the following is correct (or are they equivalent)?

(1) $$\int\phi(x)h(x)\delta(x)dx= \int\phi(x)h(a)\delta(x)dx$$ then differentiating (treating $h(a)$ as constant): as per the first comment in Why does the Dirac delta function satisfy $f(x)\delta(x-a) = f(a)\delta(x-a)$?

(2) or using the product rule: $$\int\phi(x)h(a)\delta'(x)dx+\int\phi(a)h(x)\delta'(x)dx $$

Answer 1

Since $\phi\in C_C^\infty$ and $h\in C^\infty$, then $\phi h\in C_C^\infty$. Then, we have

$$\begin{align} \langle \delta_a', \phi h\rangle&=-\langle \delta_a, (\phi h)'\rangle\\\\ &=-\langle \delta_a, \phi'h+\phi h'\rangle\\\\ &=-\phi'(a)h(a)-\phi(a)h'(a)\\\\ &=\langle \delta_a',h(a)\phi+\phi(a)h\rangle \\\\ &=\langle \delta_a',h(a)\phi\rangle +\langle \delta_a',\phi(a)h\rangle \end{align}$$