`f(x) = 3/x, c = 4` Use the alternate form of the derivative to find the derivative at x = c (if it exists)

Textbook Question

Chapter 2, 2.1 - Problem 70 - Calculus of a Single Variable (10th Edition, Ron Larson).
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leonard-chen | (Level 2) Adjunct Educator

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`lim_(x->4)(f(x) - f(c))/(x-c)`

`lim_(x->4)(3/x - 3/4)/(x-4)`

` `Simplify expression:

`lim_(x->4)((12-3x)/(4x))/(x-4)`

`lim_(x->4)(3(4-x))/(4x(x-4))`

`lim_(x->4)(-(3/(4x))) = -3/16`

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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) = 3/x

thus, f'(x) = lim h --> 0  [{(3/(x+h))  - (3/x)}}/h]

or, f'x) = lim h --> 0 [{(3x - 3(x+h)}/{h*x*(x+h)}]

or, f'(x) = lim h --> 0 [-3h/{h*x*(h+x)}] = -3/{x*(x+h)}

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

f'(x) = -3/(x^2)

Thus, f'(c) = f'(4) = -3/(4^2) = -3/16

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