$x = a (cos 2 t + 2 t sin 2 t)$ and $y = a (sin 2 t - 2 t cos 2 t)$ find second order derivative.

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Solution

$x = a (cos 2 t + 2 t sin 2 t)$
$\frac{d x}{d t} = a (- 2 sin 2 t + 2 sin 2 t + 4 t cos 2 t) = 4 a t cos 2 t$
$\frac{d^{2} x}{d t^{2}} = 4 a c o s 2 t - 8 a t S i n 2 t$

$y = a (sin 2 t - 2 t cos 2 t)$
$\frac{d y}{d t} = a (2 cos 2 t - 2 cos 2 t + 4 t sin 2 t) = 4 a t sin 2 t$
$\frac{d^{2} y}{d t^{2}} = 4 a S i n 2 t + 8 a t c o s 2 t$

So $\frac{d^{2} y}{d x^{2}} = \frac{4 S i n 2 t + 8 t C o s 2 t}{4 C o s 2 t - 8 t S i n 2 t}$

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