How do you use the limit definition to find the derivative of $f (x) = \frac{x}{x + 2}$ $f(x)=x/(x+2)$ ?

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Answer 1 · 2018-01-13T19:35:34

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

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

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

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

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

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

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

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

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

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