What is the derivative of ##y=arcsin(3x )##?

The derivative of y = arcsin(3x) is given by

##y'=3/(sqrt(1-9x^2)##

The rule for the derivative of f(x) = arcsin(x) is

##f'(x) = 1/(sqrt(1-x^2)##

However, in this instance, we are dealing with a composite function (a function of another function of x) and we have to use the as well.

So the rule for the derivative of h(x) = arcsin f(x) is given by

##h'(x)=(f'(x))/(sqrt(1-(f(x))^2)##

In our case f(x) = 3x, so f'(x) = 3 and ##(f(x))^2=9x^2##

Substituting, we get

##h'(x)=3/(sqrt(1-9x^2)##