`f(x) = 1/x^2` Find the derivative of the function by the limit process.

Textbook Question

Chapter 2, 2.1 - Problem 22 - Calculus of a Single Variable (10th Edition, Ron Larson).
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nick-teal | High School Teacher | (Level 3) Adjunct Educator

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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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hkj1385 | (Level 1) Assistant Educator

Posted on

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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