Let there be two functions expressed in the form of a parametric variable, $y=f(t)$ and $x=g(t)$and I have find the second derivative of $y$ with respect to $x$.
To do that, I have done as shown $$\frac{d^2y}{dx^2}= \frac{d}{dt}(\frac{dy}{dt})×(\frac{dt}{dx})^2$$ $$\frac{d^2y}{dx^2} = \frac{d^2y}{dt^2} \biggm/\left(\frac{dx}{dt}\right)^2$$ But I am not getting the correct answer and I don't know what is the problem with this. I want to know if I have done something wrong in the above procedure?
The first expression of yours is wrong
$\displaystyle\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{d}{dx}(\frac{\frac{dy}{dt}}{\frac{dx}{dt}})=\frac{d}{dt}(\frac{\frac{dy}{dt}}{\frac{dx}{dt}})\frac{dt}{dx}$ which on evaluating by quotient rule gives
$$\displaystyle\frac{\frac{dx}{dt}.\frac{d^2y}{dt^2}-\frac{dy}{dt}\frac{d^2x}{dt^2}}{(\frac{dx}{dt})^3}$$