math Evaluate the derivative of the expression (2-x)^square root x.

Expert Answers

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We have to find y' for y = (2-x)^(sqrt x)

Use natural logariths for both the sides

ln y = ln[ (2-x)^(sqrt x)]

use the property ln a^x = a*ln x

=> ln y = (sqrt x)*ln ( 2 - x)

Do implicit differentiation of both the sides

=> y'/y = -sqrt x/(2 - x) + (1/2)*ln(2 - x)/sqrt x

=> y' = y[(1/2)*ln(2 - x) - x/(2 - x)]/sqrt x

y' = [(2-x)^(sqrt x)]*[(1/2)*ln(2 - x) - x/(2 - x)]/sqrt x

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