Answer:
D. f'(2) = lim(-1/(4(x+2))
Step-by-step explanation:
You want the derivative of f(x) = 1/(x+2) at x=2 using the alternate definition of a derivative.
Alternate definition of a derivative
The alternate definition of a derivative tells you ...
[tex]\displaystyle f'(2) = \lim_{x\to2}\dfrac{f(x)-f(2)}{x-2}\\\\\\f'(2)=\lim_{x\to2}\dfrac{\dfrac{1}{x+2}-\dfrac{1}{2+2}}{x-2}=\lim_{x\to2}\dfrac{4-(x+2)}{4(x+2)(x-2)}\\\\\\f'(2)=\lim_{x\to2}\dfrac{2-x}{4(x+2)(x-2)}=\boxed{\lim_{x\to2}\left[\dfrac{-1}{4(x+2)}\right]}[/tex]
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Additional comment
You recognize this is the only answer choice with (x+2) in the denominator. The correct answer can be chosen on this basis alone.
