Final answer:
To find the first derivative of the function θ cos θ, use the product rule of differentiation. The first derivative is cos θ - θsin θ. To find the second derivative, differentiate the first derivative, which results in -θcos θ.
Step-by-step explanation:
To find the first derivative of the function H(θ) = θ cos θ, we can use the product rule of differentiation. Let's denote θ as u, and cos θ as v, then H(θ) = u * v, where u = θ and v = cos θ. The first derivative (H'(θ)) can be found using the formula (u * v)' = u'v + uv', which gives us H'(θ) = (θ)' * cos θ + θ * (cos θ)'. Differentiating θ with respect to θ gives 1, and differentiating cos θ with respect to θ gives -sin θ. Therefore, H'(θ) = 1 * cos θ + θ * (-sin θ) = cos θ - θsin θ.
To find the second derivative of H(θ), we need to differentiate H'(θ). Applying the product rule again, we get H''(θ) = (cos θ - θsin θ)' = (cos θ)' - (θsin θ)' = -sin θ - (θ * cos θ + sin θ) = -θcos θ.