`f(x) = (x - 6)^(2/3), c = 6` Use the alternate form of the derivative to find the derivative at x = c (if it exists)

Textbook Question

Chapter 2, 2.1 - Problem 71 - Calculus of a Single Variable (10th Edition, Ron Larson).
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leonard-chen's profile pic

leonard-chen | (Level 2) Adjunct Educator

Posted on

`lim_(x->6) (f(x) - f(c))/(x-c)`

`lim_(x->6) ((x-6)^(2/3))/(x-6)`

` `

Using division of exponents, it simplifies to:

`lim_(x->6) (x-6)^(-1/3) = 0`

gsarora17's profile pic

gsarora17 | (Level 2) Associate Educator

Posted on

`f(x)=(x-6)^(2/3)`

`f'(6)=lim_(h->0) (f(6+h)-f(6))/h`

`f'(6)=lim_(h->0) ((6+h-6)^(2/3)-(6-6)^(2/3))/h`

`f'(6)=lim_(h->0) h^(2/3)/h`

`f'(6)=lim_(h->0) h^(-1/3)`

`f'(6)=0`

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