Question

Find the dr/d(theta) for:r = (theta)sin(theta) + cos(theta)​

Answer 1

$r=\theta\sin\theta+\cos\theta$

$\dfrac{\mathrm dr}{\mathrm d\theta}=\dfrac{\mathrm d(\theta\sin\theta)}{\mathrm d\theta}+\dfrac{\mathrm d(\cos\theta)}{\mathrm d\theta}$

By the product rule,

$\dfrac{\mathrm d(\theta\sin\theta)}{\mathrm d\theta}=\dfrac{\mathrm d(\theta)}{\mathrm d\theta}\sin\theta+\theta\dfrac{\mathrm d(\sin\theta)}{\mathrm d\theta}=\sin\theta+\theta\cos\theta$

and

$\dfrac{\mathrm d(\cos\theta)}{\mathrm d\theta}=-\sin\theta$

So we have

$\dfrac{\mathrm dr}{\mathrm d\theta}=\sin\theta+\theta\cos\theta-\sin\theta$

$\implies\dfrac{\mathrm dr}{\mathrm d\theta}=\theta\cos\theta$