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If $\phi$ is a function of $t$, then $x$, for example, is written as $$x(t)=R\cos(\omega t)+\ell\sin(\phi(t)).$$ Applying the chain rule gives \begin{align} \frac{d}{dt}x(t)&=\frac{d}{dt}R\cos(\omega t)+\frac{d}{d\phi}\ell\sin(\phi)\frac{d}{dt}\phi(t)\\ &=-R\omega\sin(\omega t)+\ell\cos(\phi)\dot{\phi}. \end{align}