Question

If y = `sin^-1 (xsqrt(1 - x) + sqrt(x) sqrt(1 - x^2))`, then `("d"y)/("d"x)` = ______.

Options

• `(-2x)/sqrt(1 - x^2) + 1/(2sqrt(x - x^2))`

• `(-1)/sqrt(1 - x^2) - 1/(2sqrt(x - x^2))`

• `(-1)/sqrt(1 - x^2) + 1/(2sqrt(x - x^2))`

• None of these

Answer 1

If y = `sin^-1 (xsqrt(1 - x) + sqrt(x) sqrt(1 - x^2))`, then `("d"y)/("d"x)` = `(-1)/sqrt(1 - x^2) + 1/(2sqrt(x - x^2))`.

Explanation:

Putting x = sin A and `sqrt(x)` = sin B, we get

y = `sin^-1 (sin "A"sqrt(1 - sin^2"B") + sin"B"sqrt(1 - sin^2))`

= `sin^-1 (sin "A" cos "B" + sin "B" cos "A")`

= `sin^-1 [sin("A" + "B")]`

= A + B

= `sin^-1x + sin^-1 sqrt(x)`

∴ `("d"y)/("d"x) = 1/sqrt(1 + x^2) + 1/sqrt(1 - (sqrt(x))^2) * 1/(2sqrt(x))`

= `1/sqrt(1 - x^2) + 1/(2sqrt(x - x^2)`