Answer to Question #157320 in Calculus for Pia

Question #157320

Differentiate: 2arcsin(y/2)+√(4-y^2)


1
Expert's answer
2021-02-01T19:04:54-0500

Set f(y) = 2sin1(y2)+4y22 sin^{-1}(\frac {y}{2}) + \sqrt{4-y^2}

Since sin1(y)=11y2sin^{-1}(y) = \frac{1}{\sqrt{1-y^2}}

We have that f(y)=2(12)(11(y2)2)+12(2y)(14y2)=24y2y4y2f'(y) = 2(\frac {1}{2})(\frac {1}{\sqrt {1- (\frac {y}{2})^2}}) + \frac {1}{2}(-2y)(\frac {1}{\sqrt {4- y^2}}) = \frac {2}{\sqrt {4- y^2}} - \frac{y}{\sqrt {4- y^2}}

f(y)=2y4y2=2y2+yf'(y) = \frac {2-y}{\sqrt {4- y^2}} = \frac {\sqrt {2-y}}{\sqrt {2+y}}


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