def of derivative
#f'(x)=limh->0(f(x+h)-f(x))/(h)#
Substitution
#f'(x)=limh->0(2/(x+h+4)-2/(x+4))/(h)#
Common Denominator
#f'(x)=limh->0((2(x+4))/((x+4)(x+h+4))-(2(x+h+4))/((x+4)(x+h+4)))/(h)#
Distribute and write as a single numerator
#f'(x)=limh->0((2x+8)/((x+4)(x+h+4))-(2x+2h+8)/((x+4)(x+h+4)))/(h)#
#f'(x)=limh->0((2x+8-2x-2h-8)/((x+4)(x+h+4)))/(h)#
Simplify
#f'(x)=limh->0((cancel(2x)cancel(+8)cancel(-2x)-2hcancel(-8))/((x+4)(x+h+4)))/(h)#
#f'(x)=limh->0((-2h)/((x+4)(x+h+4)))/(h)#
Multiply by the reciprocal
#f'(x)=limh->0(-2h)/((x+4)(x+h+4))*(1/h)#
#f'(x)=limh->0(-2h)/(h(x+4)(x+h+4))#
Simplify
#f'(x)=limh->0(-2cancelh)/(cancelh(x+4)(x+h+4))#
#f'(x)=limh->0(-2)/((x+4)(x+h+4))#
Now we can substitute in a 0 for h
#f'(x)=(-2)/((x+4)(x+0+4))#
Simplify
#f'(x)=(-2)/((x+4)(x+4))#
Simplify
#f'(x)=(-2)/((x+4)^2)#
Watch this tutorial to see a similar question solved used the same methods.