\displaystyle 1.\ \text{If } y=f(x),\ \text{then }\frac{d}{dx}\left(\frac{dy}{dx}\right)\ \text{is called second order derivative of } y \text{ with respect to } x \text{ and is}
\displaystyle \text{denoted by } \frac{d^2y}{dx^2} \text{ or } y_2 \text{ or } y''. \text{ Similarly, third and higher order derivatives are defined.}

\displaystyle 2.\ \text{If } x=f(t) \text{ and } y=g(t), \text{ then}
\displaystyle \frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{g'(t)}{f'(t)}\right)
\displaystyle \text{or, } \frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{g'(t)}{f'(t)}\right)\cdot\frac{dt}{dx} \text{ or, }
\displaystyle \frac{d^2y}{dx^2}=\frac{f'(t)g''(t)-g'(t)f''(t)}{\left(f'(t)\right)^3}

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