Answer to Question #157320 in Calculus for Pia

Question

Differentiate: 2arcsin(y/2)+√(4-y^2)

Answer 1

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

Since $sin^{-1}(y) = \frac{1}{\sqrt{1-y^2}}$

We have that $f'(y) = 2(\frac {1}{2})(\frac {1}{\sqrt {1- (\frac {y}{2})^2}}) + \frac {1}{2}(-2y)(\frac {1}{\sqrt {4- y^2}}) = \frac {2}{\sqrt {4- y^2}} - \frac{y}{\sqrt {4- y^2}}$

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