CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY2 Marks
Question
If $x = at^2, y = 2$ at, then find $\frac{\text{d}^2\text{y}}{\text{dx}^2}.$
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Answer
Given,
$x = at^2$ and $y = 2$ at
On differentiating both sides w.r.t.t, we get,
$\frac{\text{dx}}{\text{dt}}=2\text{at and}\frac{\text{dy}}{\text{dt}}=2\text{a}$
Therefore,
$\frac{\text{dy}}{\text{dx}}=\frac{2\text{a}}{2\text{at}}=\frac{1}{\text{t}}$
Now, $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}\Big(\frac{\text{dy}}{\text{dx}}\Big)\times\frac{\text{dt}}{\text{dx}}$
$=\frac{\text{d}}{\text{dt}}\Big(\frac{1}{\text{t}}\Big)\times\frac{1}{2\text{at}}=-\frac{1}{\text{t}^2}\times\frac{1}{\text{2at}}=-\frac{1}{\text{2at}^3}$
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