The second derivative of a single valued function parametrically represented by $x=\phi \left(t\right)andy=\psi \left(t\right)$ (where $\phi \left(t\right)and\psi \left(t\right)$ are different function and ${\phi }^{prime}\left(t\right)\ne 0$ ) is given by $\frac{d^{2}y}{{dx}^2}=(\left(\frac{dx}{dt}\right)\left(\frac{d^{2}y}{{dt}^2}\right)-\left(\frac{d^{2}x}{{dt}^2}\right)\frac{\frac{dy}{dt}}{{\left(\frac{dx}{dt}\right)}^2}$$\frac{d^{2}y}{{dx}^2}=(\left(\frac{dx}{dt}\right)\left(\frac{d^{2}y}{{dt}^2}\right)-\left(\frac{d^{2}x}{{dt}^2}\right)\frac{\frac{dy}{dt}}{{\left(\frac{dx}{dt}\right)}^3}$$\frac{d^{2}y}{{dx}^2}=(\left(\frac{d^{2}x}{dt}\right)\left(\frac{d^y}{{dt}^2}\right)-\left(\frac{d^x}{{dt}^{}}\right)\frac{\frac{d^{2}y}{{dt}^2}}{{\left(\frac{dx}{dt}\right)}^3}$$\frac{d^{2}y}{{dx}^2}=(\left(\frac{d^{2}x}{{dt}^2}\right)\left(\frac{d^y}{{dt}^{}}\right)-\left(\frac{d^{2}x}{{dt}^2}\right)\frac{\frac{d^y}{{dt}^{}}}{{\left(\frac{dx}{dt}\right)}^3}$

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