Final answer:
To find the derivative of h(θ) = θ² sin θ, we use the product rule to get h'(θ) = 2θ sin θ + θ² cos θ, which is option a) is correct.
Step-by-step explanation:
To differentiate the function h(θ) = θ² sin θ, we will use the product rule from calculus. The product rule states that if you have a function defined as the product of two functions, say u(θ) and v(θ), then the derivative of this function with respect to θ is u'v + uv'. Applying this to our function, we let u(θ) = θ² and v(θ) = sin θ. Then, we find the derivatives of u and v separately, which are u'(θ) = 2θ and v'(θ) = cos θ respectively.
Now the derivative of h(θ) is given by u'v + uv' which is 2θ sin θ + θ² cos θ. Therefore, the correct answer is h'(θ) = 2θ sin θ + θ² cos θ, which corresponds to option a).
To differentiate h(θ) = θ² sin θ, we can use the product rule of differentiation.
Let's break down the function into two parts:
Part 1: θ²
Part 2: sin θ
Using the product rule, the differentiated function is h'(θ) = Part 1 * derivative of Part 2 + derivative of Part 1 * Part 2.
When we differentiate Part 1 and Part 2, we get:
Derivative of Part 1 = 2θ
Derivative of Part 2 = cos θ
Plugging these values back into the product rule, we have:
h'(θ) = θ² * cos θ + 2θ * sin θ
Therefore, the correct option is a. h'(θ) = 2θ sin θ + θ² cos θ.