Question

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

Answer 1

Please see the explanation

Explanation:

The limit definition is:

`lim_(hto0)(f(x + h) - f(x))/h`

We compute `f(x + h)`:

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

Substitute `f(x + h) and f(x)` into the definition:

`lim_(hto0)((x + h)/(x + h + 4) - (x)/(x + 4))/h`

Multiply the expression by 1 in the form of `((x + h + 4)(x + 4))/((x + h + 4)(x + 4))`

`lim_(hto0)((x + h)/(x + h + 4) - (x)/(x + 4))/h((x + h + 4)(x + 4))/((x + h + 4)(x + 4)) =`

`lim_(hto0)(((x + h)(x + h + 4)(x + 4))/(x + h + 4) - (x(x + h + 4)(x + 4))/(x + 4))/(h(x + h + 4)(x + 4)) =`

`lim_(hto0)((x + h)(x + 4) - (x(x + h + 4)))/(h(x + h + 4)(x + 4)) =`

`lim_(hto0)((x^2 + 4x + hx + 4h) - (x^2 + hx + 4x))/(h(x + h + 4)(x + 4)) =`

`lim_(hto0)(4h)/(h(x + h + 4)(x + 4)) =`

`lim_(hto0)4/((x + h + 4)(x + 4))`

Now it is ok to let h become 0:

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

`4/(x + 4)^2`