Question

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

Answer 1

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

Explanation:

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.