Question

How do you find the derivative of `arcsin(5x)`?

Answer 1

`y'=5/sqrt(1-(5x)^2)`

Explanation:

`y=sin^-1(5x)`

Substitute for `x`

`sinu=5x`

`cosu*du=5dx`

`(du)/dx=5/cosu` `color(green)(rarr(1)`

After substitution,

`y=sin^-1sinu`

`color(blue)(sin^-1sinu=u`

`y=u`

Differentiate with respect to `x`

`y'=(du)/dx`

Substitute from `color(green) ((1)`

`y'=5/cosu`

`y'=5/sqrt(1-sin^2u)`

Reverse the substitution

`y'=5/sqrt(1-(5x)^2)`

`color(red) "and the general formula to find the derivative of arcsin functions"`

`color(green)(d/dxsin^-1u=1/sqrt(1-u^2)(du)/dx`