`g(x) = (x + 3)^(1/3), c = -3` Use the alternate form of the derivative to find the derivative at x = c (if it exists)

Textbook Question

Chapter 2, 2.1 - Problem 72 - 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->-3) (g(x) - g(c))/(x-c)`

`lim_(x->-3) ((x+3)^(1/3))/(x+3)`

Using division of exponents, it simplifies to:

`lim_(x->-3) (x+3)^(-2/3) = 0`

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gsarora17's profile pic

gsarora17 | (Level 2) Associate Educator

Posted on

`g(x)=(x+3)^(1/3)`

`g'(-3)=lim_(h->0) (g(-3+h)-g(-3))/h`

`g'(-3)=lim_(h->0) ((-3+h+3)^(1/3)-(-3+3)^(1/3))/h`

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

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

`g'(-3)=0`

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