Question

Differentiate the following w.r.t. x : `sin^-1((cossqrt(x) + sinsqrt(x))/sqrt(2))`

Answer 1

y = `sin^-1((cossqrt(x) + sinsqrt(x))/sqrt(2))`

= `sin^-1(1/sqrt(2)cossqrt(x) + 1/sqrt(2)sinsqrt(x))`
Put,
`(1)/sqrt(2)` = sinx

`(1)/sqrt(2)` = cosα
Also,
sin²α + cos²α = `(1/sqrt(2))^2 + (1/sqrt(2))^2` = 1
And,
tanα = 1
∴ α = tan^–11
y = `sin^(–1)(sinα.cossqrt(x) + cosα.sin(x)`
= `sin^(-1)(sin(α + sqrt(x)))`
y = `α + sqrt(x)`
y = `tan^-1(1) + sqrt(x)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(tan^-1 + sqrt(x))`

= `0 + 1/(2sqrt(x))`

= `(1)/(2sqrt(x))`.