Question

x = a(cos(∅) + ∅ * sin(∅)) , y = a(sin(∅) - ∅ * cos(∅) - cos(theta)) at ∅ = π/6 Find the derivative of (dy/dx)

Answer 1

Explanation:

Take derivative of each with respect to θ.

`x=a(cos\ \theta+\theta \ sin\ \theta)\\(dx)/(d\theta)=a(-sin\ \theta+\theta \ cos\ \theta+sin\ \theta)\\ (dx)/(d\theta)=a\ \theta \ cos\ \theta\\\\y=a(sin\ \theta-\theta\ cos\ \theta-cos\ \theta)\\(dy)/(d\theta) =a(cos\ \theta+\theta\ sin\ \theta-cos\ \theta+sin\ \theta)\\(dy)/(d\theta) =a(\theta\ sin\ \theta+sin\ \theta)\\(dy)/(d\theta) =a\ sin\ \theta \ (\theta+1)`

Divide to find dy/dx.

`(dy)/(dx) =(dy)/(d\theta) / (dx)/(d\theta) \\(dy)/(dx) =(a\ sin\ \theta \ (\theta+1)) / (a\ \theta \ cos\ \theta)\\(dy)/(dx)=tan\ \theta \ (\theta +1)/(\theta)\\(dy)/(dx)=tan\ \theta \ (1+(1)/(\theta))`

Evaluate at θ = π/6:

`(dy)/(dx)=tan\ (\pi)/(6) \ (1+(6)/(\pi))\\(dy)/(dx)=(1)/(√(3)) \ (1+(6)/(\pi))`