Question

If x = 3 tan t and y = 3 sec t, then find $\frac{d^2y}{dx^2} \, at\,t=\frac{\pi}{4}$, is:

Answer 1

We have x=3tan t and y=3sec t

∴$\frac{dx}{dt}=3sec^2t$

and $\frac{dy}{dt}=3\,sec\,t\,tan\,t$

since, $\frac{dy}{dt}=\frac{tan\,t}{sec\,t}=sin\,t$

Again differentiating w.r.t. x

$\frac{d^2y}{dx^2}=cost\,\frac{dt}{dx}$

$\frac{d^2y}{dx^2}=\frac{cos\,t}{3sec^2\,t}=\frac{cos^3t}{3}$

since, at $x=\frac{\pi}{4}$

$=\frac{1}{3\times2\sqrt2}=\frac{1}{6\sqrt2}$