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

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Chapter 2, 2.1 - Problem 21 - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385 | (Level 1) Assistant Educator

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the alternate method to find the derivative of the function is the limit form.

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-1)

thus, f'(x) = lim h --> 0  [{(1/(x+h-1)) - (1/(x-1))}/h]

or, f'x) = lim h --> 0 [{(x-1) - (x+h-1)}/{h(x-1)*(x+h-1)}]

or, f'(x) = lim h --> 0 [-h/{h*(x-1)*(x+h-1)}] = -1/{(x-1)*(x+h-1)}

putting the value of h = 0 in the above  expression we get

f'(x) = -1/{(x-1)^2)}

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