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

Expert Answers

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`lim_(x->2) (f(x) - f(c))/(x-c)`

`lim_(x->2) (x^3 + 6x - 20)/(x-2)`

Simplify using synthetic division.

`lim_(x-> 2) x^2 + 2x + 10 = 18`

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`f(x)=x^3+6x` ` `

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

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

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

`f'(2)=lim_(h->0) (8+h^3+12h+6h^2+12+6h-20)/h`

`f'(2)=lim_(h->0) (h^3+6h^2+18h)/h`

`f'(2)=lim_(h->0) h^2+6h+18`

`f'(2)=18`

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