Question

differentiate. h(\theta)=\theta^{2}\sin(\theta) h(\theta)=

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = \theta^{2}$ and $v=\sin(\theta)$.

Step2: Differentiate $u$

The derivative of $u=\theta^{2}$ with respect to $\theta$ is $u' = 2\theta$ using the power - rule $\frac{d}{d\theta}(\theta^{n})=n\theta^{n - 1}$.

Step3: Differentiate $v$

The derivative of $v = \sin(\theta)$ with respect to $\theta$ is $v'=\cos(\theta)$.

Step4: Calculate $h'(\theta)$

Using the product - rule $h'(\theta)=u'v+uv'$, we substitute $u = \theta^{2}$, $u' = 2\theta$, $v=\sin(\theta)$ and $v'=\cos(\theta)$. So $h'(\theta)=2\theta\sin(\theta)+\theta^{2}\cos(\theta)$.

Answer:

$2\theta\sin(\theta)+\theta^{2}\cos(\theta)$