How do you use the limit definition of the derivative to find the derivative of $f(x)=x/(x+1)$ ?

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Answer 1 · 2017-02-13T13:24:49

$d/dx ( x/(x+1) ) = 1/((x+1)^2$

By definition we have:

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

So:

$d/dx ( x/(x+1) ) = lim_(h->0) ( (x+h)/(x+h+1) - x/(x+1)) /h$

$d/dx ( x/(x+1) ) = lim_(h->0) 1/h(( (x+h)(x+1) - x(x+h+1))/((x+1)(x+h+1)))$

$d/dx ( x/(x+1) ) = lim_(h->0) 1/h((cancelx^2+cancel(hx)+cancelx+h - cancelx^2 -cancel(hx) -cancelx)/((x+1)(x+h+1)) )$

$d/dx ( x/(x+1) ) = lim_(h->0) 1/cancel(h)(cancel(h)/((x+1)(x+h+1)) )$

$d/dx ( x/(x+1) ) = lim_(h->0) 1/((x+1)(x+h+1))$

$d/dx ( x/(x+1) ) = 1/((x+1)^2$

How do you use the limit definition of the derivative to find the derivative of f(x)=x/(x+1)f(x)=x/(x+1)?