We will use the quotient rule:
#d/dx[f(x)/g(x)]=y'=(f'(x)*g(x)-g'(x)*f(x))/(g(x))^2#
But before that however, let's find the derivative of #arcsin(3x)#
#--------------------#
Let #y=arcsin(3x)#
Take the sines of both sides
#sin(y)=3x#
Differentiate both sides W.R.T #x#
#dy/dx*cos(y)=3#
Divide #cos(y)# to both sides
#dy/dx=3/cos(y)#
We now must rewrite in terms of #x#
Since , #color(blue)(sin(y)=(3x)/1#
Then, #color(red)(cos(y)=sqrt(1-9x^2)/1=sqrt(1-9x^2)#
#:.dy/dx=(3)/sqrt(1-9x^2)#
#--------------------#
Now finding the derivative:
Let #f(x)=arcsin(3x)# & #g(x)=x#
So #f'(x)=3/sqrt(1-9x^2)# & #g'(x)=1#
Substituting into the quotient rule we get:
#y'=(3/sqrt(1-9x^2)*x-1*arcsin(3x))/(x)^2#
Simplify:
#y'=((3x)/sqrt(1-9x^2)-arcsin(3x))/x^2#
Combine Fractions in the numerator:
#color(blue)((3x)/sqrt(1-9x^2)-arcsin(3x)*(sqrt(1-9x^2)/sqrt(1-9x^2))#
#color(blue)((3x)/sqrt(1-9x^2)-(arcsin(3x)sqrt(1-9x^2))/sqrt(1-9x^2))#
#y'=((3x-arcsin(3x)sqrt(1-9x^2))/sqrt(1-9x^2))/x^2#
Apply the fraction rule for further simplification
#y'=(3x-arcsin(3x)sqrt(1-9x^2))/sqrt(1-9x^2)*1/x^2#
#y'=(3x-arcsin(3x)sqrt(1-9x^2))/(x^2sqrt(1-9x^2))#
