Quotient Rule
f'(x)=(vu'-uv')/v^2f'(x)=vu'−uv'v2
u=3x^2+4u=3x2+4
u'=6xu'=6x
v=sqrt(1+x^2)=(1+x^2)^(1/2)v=√1+x2=(1+x2)12
v'=(1/cancel2)(1+x^2)^(-1/2)*cancel2x=x/sqrt(1+x^2)v'=(12)(1+x2)−12⋅2x=x√1+x2
f'(x)=((1+x^2)^(1/2)*6x-(3x^2+4)(x/sqrt(1+x^2)))/((1+x^2)^(1/2))^2f'(x)=(1+x2)12⋅6x−(3x2+4)(x√1+x2)((1+x2)12)2
Simplify
f'(x)=(6xsqrt(1+x^2)-((x(3x^2+4))/sqrt(1+x^2)))/(1+x^2)f'(x)=6x√1+x2−(x(3x2+4)√1+x2)1+x2
f'(x)=(6xsqrt(1+x^2)-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)f'(x)=6x√1+x2−(3x3+4x√1+x2)1+x2
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)f'(x)=6x√1+x2⋅(√1+x2√1+x2)−(3x3+4x√1+x2)1+x2
Simplify
f'(x)=(((6x(1+x^2))/(sqrt(1+x^2)))-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)f'(x)=(6x(1+x2)√1+x2)−(3x3+4x√1+x2)1+x2
Distribute
f'(x)=((6x+6x^3)/(sqrt(1+x^2))-((3x^3+4x)/sqrt(1+x^2)))/(1+x^2)f'(x)=6x+6x3√1+x2−(3x3+4x√1+x2)1+x2
Numerator simplified
f'(x)=((6x+6x^3-3x^3-4x)/sqrt(1+x^2))/(1+x^2)f'(x)=6x+6x3−3x3−4x√1+x21+x2
f'(x)=((2x+3x^3)/sqrt(1+x^2))/(1+x^2)f'(x)=2x+3x3√1+x21+x2
Multiply by the reciprocal
f'(x)=(2x+3x^3)/sqrt(1+x^2)*1/(1+x^2)f'(x)=2x+3x3√1+x2⋅11+x2
f'(x)=(2x+3x^3)/(sqrt(1+x^2)*(1+x^2))f'(x)=2x+3x3√1+x2⋅(1+x2)
Simplify
f'(x)=(2x+3x^3)/((1+x^2)^(1/2)*(1+x^2)^(2/2))f'(x)=2x+3x3(1+x2)12⋅(1+x2)22
Simplify the denominator
f'(x)=(2x+3x^3)/((1+x^2)^(3/2))f'(x)=2x+3x3(1+x2)32
Factor out an x from the numerator
f'(x)=(x(2+3x^2))/((1+x^2)^(3/2))f'(x)=x(2+3x2)(1+x2)32
Watch these examples of the quotient rule.
