Quotient Rule
#f'(x)=(vu'-uv')/v^2#
#u=3x^2+4#
#u'=6x#
#v=sqrt(1+x^2)=(1+x^2)^(1/2)#
#v'=(1/cancel2)(1+x^2)^(-1/2)*cancel2x=x/sqrt(1+x^2)#
#f'(x)=((1+x^2)^(1/2)*6x-(3x^2+4)(x/sqrt(1+x^2)))/((1+x^2)^(1/2))^2#
Simplify
#f'(x)=(6xsqrt(1+x^2)-((x(3x^2+4))/sqrt(1+x^2)))/(1+x^2)#
#f'(x)=(6xsqrt(1+x^2)-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)#
Common Denominator
#f'(x)=(6xsqrt(1+x^2)*((sqrt(1+x^2))/(sqrt(1+x^2)))-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)#
Simplify
#f'(x)=(((6x(1+x^2))/(sqrt(1+x^2)))-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)#
Distribute
#f'(x)=((6x+6x^3)/(sqrt(1+x^2))-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)#
Numerator simplified
#f'(x)=((6x+6x^3-3x^3-4x)/sqrt(1+x^2))/(1+x^2)#
#f'(x)=((2x+3x^3)/sqrt(1+x^2))/(1+x^2)#
Multiply by the reciprocal
#f'(x)=(2x+3x^3)/sqrt(1+x^2)*1/(1+x^2)#
#f'(x)=(2x+3x^3)/(sqrt(1+x^2)*(1+x^2))#
Simplify
#f'(x)=(2x+3x^3)/((1+x^2)^(1/2)*(1+x^2)^(2/2))#
Simplify the denominator
#f'(x)=(2x+3x^3)/((1+x^2)^(3/2))#
Factor out an x from the numerator
#f'(x)=(x(2+3x^2))/((1+x^2)^(3/2))#
Watch these examples of the quotient rule.