Find $\frac{d y}{d x}$ , if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

x = sin t, y = cos 2t.

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Solution

Given, x = sin t, y = cos 2t

Differentiating w.r.t. t, we get

$∴ \frac{d x}{d t} = c o s t a n d \frac{d y}{d t} = - (s i n 2 t) 2 ∴ \frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}} = \frac{d y}{d t} \times \frac{d y}{d x} (∵ \frac{d y}{d x} = \frac{d y / d t}{d x / d t}) = \frac{- 2 s i n 2 t}{c o s t} = \frac{- 2 (2 s i n t c o s t)}{c o s t} = - 4 s i n t (∵ s i n 2 θ = 2 s i n θ c o s θ)$

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