Question

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

Answer 1

`r=\theta\sin\theta+\cos\theta`

`(\mathrm dr)/(\mathrm d\theta)=(\mathrm d(\theta\sin\theta))/(\mathrm d\theta)+(\mathrm d(\cos\theta))/(\mathrm d\theta)`

By the product rule,

`(\mathrm d(\theta\sin\theta))/(\mathrm d\theta)=(\mathrm d(\theta))/(\mathrm d\theta)\sin\theta+\theta(\mathrm d(\sin\theta))/(\mathrm d\theta)=\sin\theta+\theta\cos\theta`

and

`(\mathrm d(\cos\theta))/(\mathrm d\theta)=-\sin\theta`

So we have

`(\mathrm dr)/(\mathrm d\theta)=\sin\theta+\theta\cos\theta-\sin\theta`

`\implies(\mathrm dr)/(\mathrm d\theta)=\theta\cos\theta`