Questions: Differentiate. f(θ)=θ cos(θ) sin(θ) f′(θ)=

Solution Steps

To differentiate the function \( f(\theta) = \theta \cos(\theta) \sin(\theta) \), we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions is given by \( (uv)' = u'v + uv' \). Here, we have three functions multiplied together, so we will apply the product rule iteratively. Additionally, we will use the chain rule to differentiate the trigonometric functions.

The function given is \( f(\theta) = \theta \cos(\theta) \sin(\theta) \). To differentiate this function, we apply the product rule. The product rule for three functions \( u(\theta) = \theta \), \( v(\theta) = \cos(\theta) \), and \( w(\theta) = \sin(\theta) \) is:

\[ f'(\theta) = u'vw + uv'w + uvw' \]

• The derivative of \( u(\theta) = \theta \) is \( u'(\theta) = 1 \).
• The derivative of \( v(\theta) = \cos(\theta) \) is \( v'(\theta) = -\sin(\theta) \).
• The derivative of \( w(\theta) = \sin(\theta) \) is \( w'(\theta) = \cos(\theta) \).

Substitute the derivatives into the product rule formula:

\[ f'(\theta) = (1) \cdot \cos(\theta) \cdot \sin(\theta) + \theta \cdot (-\sin(\theta)) \cdot \sin(\theta) + \theta \cdot \cos(\theta) \cdot \cos(\theta) \]

Simplify the expression:

\[ f'(\theta) = \cos(\theta) \sin(\theta) - \theta \sin^2(\theta) + \theta \cos^2(\theta) \]

Final Answer