Question

If H(θ) = θ sin θ, find H'(θ) and H''(θ)

Answer 1

To find H'(θ), use the product rule for differentiation. To find H''(θ), differentiate H'(θ) using the product rule again.

Step-by-step explanation:

To find the first derivative H'(θ) of the function H(θ) = θ sin θ, we can use the product rule for differentiation. The product rule states that if we have two functions, u(θ) and v(θ), then the derivative of their product is given by (u'v + uv'). In this case, u(θ) = θ and v(θ) = sin θ. So, we have:

H'(θ) = (θ' sin θ) + (θ cos θ) = sin θ + θ cos θ

To find the second derivative H''(θ), we can differentiate the first derivative using the product rule again:

H''(θ) = (sin θ + θ cos θ)' = (sin θ)' + (θ cos θ)' = cos θ + θ (-sin θ) = cos θ - θ sin θ