Calculus of a Single Variable, Chapter 2, 2.1, Section 2.1, Problem 22

By limit process, the derivative of a function f(x) is :-
f'(x) = lim_(h -> 0) [{f(x+h) - f(x)}/h]
Now, the given function is :-
f(x) = 1/(x^2)
thus, f'(x) = lim_(h -> 0) [{{1/(x+h)^2} - {1/(x^2)}}/h]
or, f'(x) = lim_(h -> 0) [{(x^2) - (x+h)^2}/{h*(x^2)*(x+h)^2}]
or, f'(x) = lim_(h -> 0) [{-2hx - (h^2)}/{h*(x^2)*(x+h)^2}]
or, f'(x) = lim_(h -> 0)[{-2x - h}/{(x^2)*(x+h)^2}]
putting the value of h = 0 in the above expression we get
f'(x) = -2x/(x^4) = -2/(x^3)

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