If y= ^ -1 ( 3x+4 1-x^2 5 ), find dy dx - Vidyadip

Question

$\text{If y}=\cos^{-1}\bigg(\frac{3\text{x}+4\sqrt{1-\text{x}^2}}{5}\bigg),\ \text{find}\ \frac{\text{dy}}{\text{dx}}\dot{}$

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Answer

$=\cos^{-1}\bigg[\frac{3}{5}\text{x}+\frac{4}{5}\sqrt{1-\text{x}^2}\bigg]$
$=\cos^{-1}\bigg[\frac{3}{5}\dot{}\cos\theta+\frac{4}{5}\sin\theta\bigg]\ \text{where x}=\cos\theta$
$=\cos^{-1}\big[\cos\alpha\dot{}\cos\theta+\sin\alpha\dot{}\sin\theta\big],\;\because\ \text{if}\: \frac{3}{5}=\cos\alpha,\text{then}\frac{4}{5}=\sin \ \alpha $
$=\cos^{-1}\big[\cos\big(\alpha-\theta\big)\big]=\alpha-\theta=\cos^{-1}\big(3/5\big)-\cos^{-1}\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{1-\text{x}^2}}$

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