If x=f(t) and y=g(t), then write the value of d^2y dx^2 . - Vidyadip

Question

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

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Answer

We are given
$\text{x}=\text{f}(\text{t})$
$\text{Y}=\text{g}(\text{t})$
Then $\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{dt}}\times\frac{\text{dt}}{\text{dx}}=\frac{\text{g}'(\text{t})}{\text{f}'(\text{t})}$
$\therefore\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}\frac{\text{f}'(\text{t})\text{g}'\text({t})-\text{g}'(\text{t})\text{f}''\text({t})}{[\text{f}'(\text{t})]^3}$

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