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Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation4 Marks
Question
Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\frac{\sin\text{x}+\cos\text{x}}{\sqrt{2}}\Big\},-\frac{3\pi}{4}<\text{x}<\frac{\pi}{4}$
✓

Answer

Let $\text{y}=\sin^{-1}\Big\{\frac{\sin\text{x}+\cos\text{x}}{\sqrt{2}}\Big\}$
$=\sin^{-1}\bigg\{\sin\text{x}\Big(\frac{1}{\sqrt{2}}\Big)+\cos\text{x}\Big(\frac{1}{\sqrt{2}}\Big)\bigg\}$
$=\sin^{-1}\Big\{\sin{\text{x}}\cos\frac{\pi}{4}+\cos\text{x}\times\sin\frac{\pi}{4}\Big\}$
$\text{y}=\sin^{-1}\Big\{\sin\Big(\text{x}+\frac{\pi}{4}\Big)\Big\}$
Here, $\frac{-3\pi}{4}<\text{x}<\frac{\pi}{4}$
$\Rightarrow\Big(\frac{-3\pi}{4}+\frac{\pi}{4}\Big)$
$\Big[\text{Since},\sin^{-1}(\sin\theta)=\theta, \text{ if }\theta\in\Big[\frac{-\pi}{2},\frac{\pi}{2}\Big]\Big]$
Differentiating it with respect to x,
$\frac{\text{dy}}{\text{dx}}=1+0$
$\frac{\text{dy}}{\text{dx}}=1$

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