If `x=a(cos2t+2tsin2t)` and `y=a(sin2t-2tcos2t)` , then find `(d^2y)/(dx^2)` .

Explanation

We have,
`
x=a(cos 2 t+2 t sin 2 t) and y=a(sin 2 t-2 t cos 2 t)`

On differentiating with respect to t, we get
`
frac{d x}{d t}=frac{d}{d t}[a(cos 2 t+2 t sin 2 t)]=a(-2 sin 2 t+2 sin 2 t+4 t cos 2 t) =a(4 t cos 2 t)`
and
`frac{d y}{d t}=frac{d}{d t}[a(sin 2 t-2 t cos 2 t)]=a(2 cos 2 t-2 cos 2 t+4 t sin 2 t) =a(4 t sin 2 t)`

Now, `(frac{d y}{d x})=frac{frac{d y}{d t}}{frac{d x}{d t}}=frac{a(4 t sin 2 t)}{a(4 t cos 2 t)}=tan 2 t`

Therefore,


`frac{d^{2} y}{d x^{2}}=frac{d}{d x}(frac{d y}{d x})=frac{d}{d x}(tan 2 t) `
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