How do you use the definition of a derivative to find the derivative of #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 #