Come si trova la derivata di #f (x) = 1 / x ^ 2 # usando il processo limite?

By definition of the derivative # f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h #
Quindi con # f(x) = 1/x^2 # we have;

# f'(x)=lim_(h rarr 0) ( 1/(x+h)^2 -1/x^2 ) / h #
# :. f'(x)=lim_(h rarr 0)1/h*(x^2-(x+h)^2)/((x+h)^2x^2) #
# :. f'(x)=lim_(h rarr 0) (x^2-(x^2+2hx+h^2))/(h(x+h)^2x^2) #
# :. f'(x)=lim_(h rarr 0) (x^2-x^2-2hx-h^2)/(h(x+h)^2x^2) #
# :. f'(x)=lim_(h rarr 0) (-2x-h)/((x+h)^2x^2) #
# :. f'(x)=(-2x)/((x)^2x^2) #
# :. f'(x)=-2/x^3 #