How do you use the definition of a derivative to find the derivative of #f(x) = 7/(9x)#?

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Answer 1 · 2016-03-07T16:21:44

For all #x_0 in RR "\" {0} : f'(x_0) = -7/(9x_0^2)#.
See answer below.

For all #x_0 in RR "\" {0} :#

#f'(x_0) = lim_(x->x_0)(f(x)-f(x_0))/(x-x_0)#

#= lim_(x->x_0)((7/(9x))-(7/(9x_0)))/(x-x_0)#

#= lim_(x->x_0)((7x_0-7x)/(9x*x_0))/(x-x_0)#

#= lim_(x->x_0)(-7(x-x_0))/((9x*x_0)*(x-x_0))#

#= lim_(x->x_0)-7/(9x*x_0)#

#=-7/(9x_0^2).#