How do you use the limit definition to find the derivative of #f(x)=2/(x+4)#?

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Answer 1 · 2016-10-21T03:01:16

#f'(x)=(-2)/((x+4)^2)#

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)#

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