Question

Which of the following corresponds to the derivative of f(x)=1/x+2 at x=2, using the alternate definition of a derivative, reduced to its simplest form before taking the limit

Answer 1

D. f'(2) = lim(-1/(4(x+2))

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 ...

`\displaystyle f'(2) = \lim_(x\to2)(f(x)-f(2))/(x-2)\\\\\\f'(2)=\lim_(x\to2)((1)/(x+2)-(1)/(2+2))/(x-2)=\lim_(x\to2)(4-(x+2))/(4(x+2)(x-2))\\\\\\f'(2)=\lim_(x\to2)(2-x)/(4(x+2)(x-2))=\boxed{\lim_(x\to2)\left[(-1)/(4(x+2))\right]}`

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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.