Question

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

Answer 1

$x=3\tan t$

Differentiating w.r.t. $t$, we get

$\frac{dx}{dt}=3{\sec }^{2}t$

Also,

$y=3\sec t$

Differentiating w.r.t. $t$, we get

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

Now,

$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{3\sec t\tan t}{3{\sec }^{2}t}$

$\Rightarrow \frac{dy}{dx}=\sin t$

Now, differentiating w.r.t. $x$, we have

$\frac{d^{2}y}{d{x}^2}=\cos t\frac{dt}{dx}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\cos t\left(\frac{1}{3{\sec }^{2}t}\right)=\frac{{\cos }^{3}t}{3}$

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

$\frac{d^{2}y}{d{x}^2}=\frac{{\cos }^3\frac{\pi }{4}}{3}=\frac{1}{2\sqrt{2}\times 3}=\frac{1}{6\sqrt{2}}=\frac{\sqrt{2}}{12}$