Mathematics - Differentiation Question with Solution | TestHub

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

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

$\Rightarrow \frac{dy}{dx}=\frac{{\displaystyle \raisebox{1ex}{$dy$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}}{{\displaystyle \raisebox{1ex}{$dx$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}}$

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

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\cos t·\frac{dt}{dx}=\cos t ·\frac{{\cos }^{2}t}{3}=\frac{{\cos }^{3}t}{3}$

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

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