Questions: Differentiate. h(θ)=θ^2 sin (θ) h'(θ)=

Solution Steps

To differentiate the function \( h(\theta) = \theta^2 \sin(\theta) \), we will use the product rule. The product rule states that if you have a function that is the product of two functions, \( u(\theta) \) and \( v(\theta) \), then the derivative is given by \( u'(\theta)v(\theta) + u(\theta)v'(\theta) \). Here, let \( u(\theta) = \theta^2 \) and \( v(\theta) = \sin(\theta) \).

We are given the function \( h(\theta) = \theta^2 \sin(\theta) \). To differentiate this function, we apply the product rule. The product rule states that if \( h(\theta) = u(\theta) v(\theta) \), then \( h'(\theta) = u'(\theta) v(\theta) + u(\theta) v'(\theta) \).

Let \( u(\theta) = \theta^2 \) and \( v(\theta) = \sin(\theta) \).

• The derivative of \( u(\theta) = \theta^2 \) is \( u'(\theta) = 2\theta \).
• The derivative of \( v(\theta) = \sin(\theta) \) is \( v'(\theta) = \cos(\theta) \).

Using the product rule, we find:

\[ h'(\theta) = u'(\theta) v(\theta) + u(\theta) v'(\theta) = 2\theta \sin(\theta) + \theta^2 \cos(\theta) \]

Final Answer