worked example: Differentiate f(x)=x+12x
f′(x)=(x+1)2(2)(x+1)−(2x)(1)
f′(x)=(x+1)22x+2−2x
f′(x)=(x+1)22
worked example: Differentiate f(x)=3x−1x2
f′(x)=(3x−1)2(2x)(3x−1)−(x2)(3)
f′(x)=(3x−1)26x2−2x−3x2
f′(x)=(3x−1)23x2−2x
worked example: Differentiate f(x)=2e2x4e2x−2ex+1
f′(x)=(2e2x)2(8e2x−2ex)(2e2x)−(4e2x−2ex)(4e2x)
f′(x)=2e4x16e4x−4e3x−16e4x+8e3x
f′(x)=e4x2e3x
worked example: Differentiate f(x)=x2+1(x−1)2
f′(x)=(x2+1)2(2x−2)(x2+1)−(x−1)2⋅(2x)