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) = "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}}"


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