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Question
If x and y are connected parametrically by the equation without eliminating the parameter, find `dy/dx`.
x = cos θ – cos 2θ, y = sin θ – sin 2θ
Solution
Given, x `= cos theta - cos 2 theta` and `y = sin theta - sin 2 theta`
x `= cos theta - cos 2 theta`
Differentiating both sides with respect to θ,
`dx/(d theta) = - sin theta - (- sin 2 theta) d/(d theta) (2 theta) = - sin theta + 2 sin 2 theta`
and y = `sin theta - sin 2 theta)`
`therefore dy/(d theta) = cos theta - cos 2 theta d/(d"theta) (2 theta) = cos theta - 2 cos 2 theta`
Hence, `dy/dx = (dy/(d theta))/(dx/(d theta))`
`= (cos theta - 2 cos 2 theta)/(- sin theta + 2 sin 2 theta)`
`= (cos theta - 2 cos 2 theta)/(- (sin theta - 2 sin 2 theta))`
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