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find dy/dx y= cos^-1(( sqrt(1+x) + s qrt(1-x)) /2)

 
find dy/dx 
y=cos^-1((sqrt(1+x) + sqrt(1-x)) /2)

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:
Hello Student,
Please find answer to your question below

y = cos^{-1}(\frac{\sqrt{1+x}+\sqrt{1-x}}{2})
Apply chain rule, we have
\frac{dy}{dx} = \frac{dcos^{-1}u}{dx}.\frac{du}{dx}
u = \frac{\sqrt{1+x}+\sqrt{1-x}}{2}
u^{2} = \frac{2+2\sqrt{1-x^{2}}}{4}
u^{2} =\frac{1+\sqrt{1-x^{2}}}{2}

\frac{dy}{dx} = \frac{-1}{\sqrt{1-u^{2}}}.\frac{du}{dx}
\frac{dy}{dx} = \frac{-\sqrt{2}}{\sqrt{1-\sqrt{1-x^{2}}}}.\frac{1}{2}.(\frac{1}{2\sqrt{x+1}}+\frac{-1}{2\sqrt{1-x}})
\frac{dy}{dx} = \frac{-\sqrt{2}}{\sqrt{1-\sqrt{1-x^{2}}}}.\frac{1}{4}.(\frac{1}{\sqrt{x+1}}+\frac{-1}{\sqrt{1-x}})\frac{dy}{dx} = \frac{-\sqrt{2}}{\sqrt{1-\sqrt{1-x^{2}}}}.\frac{1}{4}.(\frac{\sqrt{1-x}-\sqrt{1+x}}{\sqrt{1-x^{2}}})

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