Question

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

Answer 1

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

Explanation:

Given `f(x)=x/(x+2)`

By the limit definition of the derivative
`f'(x)=lim_(hrarr0) (f(x+h)-f(x))/h`

`color(white)("XXX")=lim_(hrarr0) ((x+h)/(x+h+2)-x/(x+2))/h`

`color(white)("XXX")=lim_(hrarr0) (((x+h) * (x+2) -x * (x+h+2))/((x+h+2) * (x+2)))/h`

`color(white)("XXX")=lim_(hrarr0)(((x^2+2h+xh+2x)-(x^2+xh+2x))/(x^2+xh+2x+2h+4))/h`

`color(white)("XXX")=lim_(hrarr0) ((2h)/(x^2+2h+xh+4x+4))/h`

`color(white)("XXX")=lim_(hrarr0)2/(x^2+2h+xh+4x+4)`

`color(white)("XXX")=2/(x^2+4x+4)color(white)("xx")orcolor(white)("xx")2/((x+2)^2)`