Question

How do you differentiate `arcsin(sqrtx)`?

Answer 1

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

Explanation:

Let `color(green)(g(x)=sqrt(x))` and `f(x)=arcsinx`
Then`color(blue)(f(color(green)(g(x)))=arcsinsqrtx)`

Since the given function is a composite function we should differentiate using chain rule.

`color(red)(f(g(x))')=color(red)(f')(color(green)(g(x)))*color(red)(g'(x))`

Let us compute `color(red)(f'(color(green)(g(x)))) and color(red)(g'(x))`

`f(x)=arcsinx`
`f'(x)=1/(sqrt(1-x^2))`
`color(red)(f'(color(green)(g(x)))=1/(sqrt(1-color(green)(g(x))^2))`
`f'(color(green)(g(x)))=1/(sqrt(1-color(green)(sqrtx)^2))`
`color(red)(f'(g(x))=1/(sqrt(1-x)))`

`color(red)(g'(x))=?`

`color(green)(g(x)=sqrtx)`
`color(red)(g'(x)=1/(2sqrtx))`

`color(red)(f(g(x))')=color(red)(f'(g(x)))*color(red)(g'(x))`
`color(red)(f(g(x))')=1/(sqrt(1-x))*1/(2sqrtx)`
`color(red)(f(g(x))')=1/(2sqrt(x(1-x)))`

Therefore,
`color(blue)((arcsinsqrtx)'=1/(2sqrt(x(1-x)))`