y=f(x)=(4+x)/(4-x).
If x increases by a small amount h then y increases by a small amount k. dy/dx approximates to k/h.
k=f(x+h)-f(x)=(4+x+h)/(4-x-h)-(4+x)/(4-x)=
((4+x+h)(4-x)-(4+x)(4-x-h))/((4-x-h)(4-x))=
(16-x²+h(4-x)-(16-x²-h(4+x))/((4-x)²-h(4-x))=
(4h-xh+4h+xh)/((4-x)²-h(4-x))=
8h/((4-x)²-h(4-x)). In the denominator the term h(4-x) is small compared to (4-x)², so we can ignore it.
Therefore k=8h/(4-x)² and k/h=8/(4-x)².
The derivative of f(x)=f'(x)=(4-x+(4+x))/(4-x)²=8/(4-x)². This is the same as k/h=dy/dx.