What's the value of the integral $\int_{-4}^{4}(\frac{\mathrm{d}}{\mathrm{d} t} \delta(2t))\sin(t)\mathrm{d} t$ ..where $\delta(t)$ is the Dirac delta

Question

Can it be solved like $$\begin{align}\int_{-4}^{4}\frac{\mathrm{d}}{\mathrm{d} t} \delta(2t)\sin(2t)\mathrm{d}t &=\frac{1}{2} \frac{\mathrm{d} }{\mathrm{d} t}\int_{-4}^{4}\delta(t)\sin(2t)\mathrm{d}t\quad ^*\\ &=\frac{1}{2} \frac{\mathrm{d} }{\mathrm{d}t} = 0\quad ^{**}\\ \end{align}$$ $^*$ Since linear operations can be interchanged.

$^{**}$ Since $\int_{-4}^{4}\delta(t)\sin(2t)\mathrm{d} t= \sin 0=0$.