[tex]r=\theta\sin\theta+\cos\theta[/tex]
[tex]\dfrac{\mathrm dr}{\mathrm d\theta}=\dfrac{\mathrm d(\theta\sin\theta)}{\mathrm d\theta}+\dfrac{\mathrm d(\cos\theta)}{\mathrm d\theta}[/tex]
By the product rule,
[tex]\dfrac{\mathrm d(\theta\sin\theta)}{\mathrm d\theta}=\dfrac{\mathrm d(\theta)}{\mathrm d\theta}\sin\theta+\theta\dfrac{\mathrm d(\sin\theta)}{\mathrm d\theta}=\sin\theta+\theta\cos\theta[/tex]
and
[tex]\dfrac{\mathrm d(\cos\theta)}{\mathrm d\theta}=-\sin\theta[/tex]
So we have
[tex]\dfrac{\mathrm dr}{\mathrm d\theta}=\sin\theta+\theta\cos\theta-\sin\theta[/tex]
[tex]\implies\dfrac{\mathrm dr}{\mathrm d\theta}=\theta\cos\theta[/tex]
