If y = ^ -1 [ 5x + 12 1 - x^ 2 13 ], find dy dx . - Vidyadip

Question

If $y = \sin^{-1 } \Bigg[\frac{\text{5x + 12}\sqrt{1 - \text{x}^{2}}}{13}\Bigg],$ find $\frac{\text{dy}}{\text{dx}}.$

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Answer

Let $\frac{5}{13} = \cos \alpha$ and $x = \sin\theta$
$\therefore\sin\alpha=\frac{12}{13},\ \cos\theta=\sqrt{1-\text{x}^{2}}$
$\therefore\text{y}=\sin^{-1}(\sin\theta\cos\alpha+\cos\theta\sin\alpha)=\sin^{-1}[\sin(\theta+\alpha)]$
$=\theta+\alpha=\sin^{-1}\text{x}\ +$ constant
$\therefore\text{dy}/\text{dx}=\frac{1}{\sqrt{1-\text{x}^{2}}}$.

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