If x = 3 t and y = 3 t, then the value of d ^ 2 y d x ^ 2 at t =...

Given $x = 3\tan t$ and $y = 3\sec t$

$⇒ \dfrac{dx}{dt} = 3\sec^{2}t$ and $\dfrac{dy}{dt} = 3\sec t \tan t$

$⇒ \dfrac{dy}{dx} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$

$⇒ \dfrac{dy}{dx} = \sin t$

$⇒ \dfrac{{d}^{2}y}{d{x}^{2}} = \cos t \cdot \dfrac{dt}{dx} = \cos t \cdot \dfrac{\cos^{2}t}{3} = \dfrac{\cos^{3}t}{3}$

At $t = \dfrac{\pi}{4}$

$\dfrac{{d}^{2}y}{d{x}^{2}} = \dfrac{1}{6\sqrt{2}}$