Question

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

Answer 1

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

Explanation:

Limit definition of a derivative is:

`f'(x) = lim_(h->0) (f(x+h) - f(x))/h`

`f(x+h) = (x+h)/(x+h+4)`

`f(x) = x/(x+4)`

`f'(x) = lim_(h->0) ((x+h)/(x+h+4) - x/(x+4))/h`

Need to get a common denominator so:

`f'(x) = lim_(h->0) ((x+h)(x+4) - x(x+h+4))/((x+4)(x+h+4)) * 1/h`

`= lim_(h->0) (x^2 + (4+h)x + 4h - x^2 -(4+h)x)/((x+4)(x+h+4)) * 1/h`

A lot of the numerator cancels:

`= lim_(h->0) (cancel(x^2) + cancel((4+h)x) + 4h - cancel(x^2) -cancel((4+h)x))/((x+4)(x+h+4)) * 1/h`

We are left with:

`= lim_(h->0) 4/((x+4)(x+h+4)) = 4/(x+4)^2`