If 2 3 2 and y= ^ -1 ( ), find dy dx . - Vidyadip

Question

If $\frac{\pi}{2}\leq\text{x}\leq\frac{3\pi}{2}$ and $\text{y}=\sin^{-1}(\sin\text{x}),$ find $\frac{\text{dy}}{\text{dx}}.$

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Answer

Here,
$\text{y}=\sin^{-1}(\sin\text{x}),\text{x}\in\Big[\frac{\pi}{2},\frac{3\pi}{1}\Big]$
$\Big[\text{Since},\sin^{-1}(\sin\text{x})=\text{x},\text{if x}\in\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\Big]$
$\text{y}=\pi-\text{x}$
Differentiating it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\pi-\text{x})$
$0-1$
$\frac{\text{dy}}{\text{dx}}=-1$

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