Differentiate. the following

Question

Differentiate.

f(θ) = θ cos(θ) sin(θ)

Answer 1

f(T) = T * [ cos T sin T ]

= T * g(T) where g(T) = cos T * sin T

g'(T) = (cosT)^2 - (sin T)^2

f'(T) = T * g'(T) + g(T) * dT

= T [ (cosT)^2 - (sin T)^2] + (cos T sin T) dT

Answer 2

This problem asks you to apply your knowledge of the product rule. Remember the product rule for the derivative of a product of three functions:

[u v w]' = (u')vw + u(v')w + uv(w')

Now we apply this rule to the function given in the problem: f(θ) = θcos(θ)sin(θ). Usually we have x or y as our variable, but here we have theta, θ. Don't let this guy give you any trouble. We deal with him the same way we deal with x and y. If it helps, you may temporarily use x instead of θ while working through the problem. Just remember to put the θ back before you turn in the problem.

f'(θ) = 1 * cos(θ)sin(θ) + θ(-sin(θ))(sin(θ)) + θ(cos(θ))(cos(θ))

Simplify:

f'(θ) = cos(θ)sin(θ) - θsin²(θ) + θcos²(x)