`f(x) = x^3 + 2x^2 + 1, c = -2` Use the alternate form of the derivative to find the derivative at x = c (if it exists)

Textbook Question

Chapter 2, 2.1 - Problem 67 - 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->-2) (f(x) - f(c))/(x-c)`

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

Simplify using synthetic division.
` `

`lim_(x->-2) x^2 = 4`

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gsarora17 | (Level 2) Associate Educator

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

`f'(-2)=lim_(h->0) (((-2+h)^3+2(-2+h)^2+1) -((-2)^3+2(-2)^2+1))/h`

`f'(-2)=lim_(h->0) (((-2)^3+h^3+3(-2h)(-2+h)+2(4+h^2-4h)+1)-(-8+8+1))/h`

`f'(-2)=lim_(h->0) ((-8+h^3+12h-6h^2+8+2h^2-8h+1)-1)/h`

`f'(-2)=lim_(h->0) (h^3-4h^2+4h)/h`

`f'(-2)=lim_(h->0) h^2-4h+4`

`f'(-2)=4`

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