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Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives2 Marks
Question
If $\text{x}=2\text{ at},\text{y}=\text{at}^2,$ where a is a constant, then find $\frac{\text{d}^2\text{y}}{\text{dx}^2}\text{ at}\text{ x}=\frac{1}{2}.$
✓

Answer

Here,
$\text{x}=2\text{at}\ \text{and}\ \text{y}=\text{at}^2$
Differentiating w.r.t.t, we get
$\frac{\text{dx}}{\text{dt}}=2\text{a}\ \text{and}\ \frac{\text{dy}}{\text{dt}}=2\text{at}$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{2\text{at}}{2\text{a}}=\text{t}$
Differentiating w.r.t.t, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=1\times\frac{\text{dt}}{\text{dx}}=\frac{1}{2\text{a}}$
Now $\Big[\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big]_{\text{x}=\frac{1}{2}}=\frac{1}{2\text{a}}$

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