What are the Fourier coefficients of multiplying a function with the derivative of another function?

Question

Consider a Fourier series $x(t)=\sum_{n=-\infty}^{\infty} C_n e^{j n \omega_0 t}$. Let's assume $\mathrm{x}_1(\mathrm{t}) {\leftrightarrow} \mathrm{C}_{\mathrm{n}}, \mathrm{x}_2(\mathrm{t}){\leftrightarrow} \mathrm{D}_{\mathrm{n}}$. Then we know that $\mathrm{x}_1(\mathrm{t}) \cdot \mathrm{x}_2(\mathrm{t}) {\leftrightarrow} \sum_{\mathrm{k}=-\infty}^{\infty} \mathrm{C}_{\mathrm{k}} \mathrm{D}_{\mathrm{n}-\mathrm{k}}$. But how about $\mathrm{x}_1(\mathrm{t}) \cdot \frac{d}{dt}\mathrm{x}_2(\mathrm{t}) $ ? Is it $\mathrm{x}_1(\mathrm{t}) \cdot \frac{d}{dt}\mathrm{x}_2(\mathrm{t}) {\leftrightarrow} \sum_{\mathrm{k}=-\infty}^{\infty} \mathrm{C}_{\mathrm{k}}.(in\omega_0). \mathrm{D}_{\mathrm{n}-\mathrm{k}}$ ?