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