Question

What is the derivative of `f(x)=arcsin sqrt sinx`?

Answer 1

One can derive the derivative for `arcsinx` with implicit differentiation if it is not easy to remember it.

`y = arcsinx`
`siny = x`
`cosy((dy)/(dx)) = 1`
`(dy)/(dx) = 1/(cosy) = 1/(sqrt(1-sin^2y)) = 1/(sqrt(1-x^2))`

since `sin^2x + cos^2x = 1`.

Thus, take this further with the Chain Rule.

`d/(dx)[arcsinsqrt(sinx)] = 1/(sqrt(1-(sqrtsinx)^2)) * 1/((2sqrtsinx)) * cosx`

`= color(blue)(cosx/(2sqrtsinxsqrt(1-sinx)))`