Find the indicated nth derivative of the following: write the solution of each number

Question

7. 2nd derivative of y=xArcsinx + √1-x²

Answer 1

Product rule for 1st term and write 2nd term as (1 - x²)^1/2 before taking derivative:

y' = x [1/√(1 - x²)] + sin^-1(x) + (1/2) (1 - x²)^-1/2(-2x)

y' = sin^-1(x) + x/√(1- x²) - x/√(1 - x²) = sin^-1(x)

y'' = 1/√(1 - x²)

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