How do you use the definition of a derivative to find the derivative of $f (x) = \frac{1}{x}$ $f(x)=1/x$ ?

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Answer 1 · 2018-03-04T11:53:20

$f'(x) = -1/x^2$

The definition of a derivative is:

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

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

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

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

$=> lim_(h to 0 ) (-cancel(h)/(x(x+h) ) )/cancel(h)$

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

As $h to 0 " " x(x+h) to x^2$

$=> -1/x^2$

$therefore f'(x) = -1/x^2$

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