If the solve the problem

Question:

If $x=f(t)$ and $y=g(t)$, then write the value of $\frac{d^{2} y}{d x^{2}}$

Solution:

Given:

$x=f(t)$ and $y=g(t)$

$\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{f}^{\prime}(\mathrm{t}) ; \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{g}^{\prime}(\mathrm{t})$

$\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{g^{\prime}(t)}{f^{\prime}(t)}$

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\frac{\mathrm{dx}}{\mathrm{dt}}}$

$=\frac{1}{f^{\prime}(t)}\left\{\frac{1}{f^{\prime}(t)^{2}}\left(g^{\prime \prime}(t) f^{\prime}(t)-f^{\prime \prime}(t) g^{\prime}(t)\right)\right\}$

$=\frac{\left(\mathrm{g}^{\prime \prime}(\mathrm{t}) \mathrm{f}^{\prime}(\mathrm{t})-\mathrm{f}^{\prime \prime}(\mathrm{t}) \mathrm{g}^{\prime}(\mathrm{t})\right)}{\left(\mathrm{f}^{\prime}(\mathrm{t})\right)^{3}}$

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