Question

Find the derivative of f(x) = 6/x at x = -2. (2 points)

Answer 1

Using the definition of the derivative,

$\displaystyle f'(-2) = \lim_{x\to-2}\frac{f(x) - f(-2)}{x - (-2)}$

$\displaystyle f'(-2) = \lim_{x\to-2}\frac{\dfrac6x + 3}{x + 2}$

$\displaystyle f'(-2) = \lim_{x\to-2}\frac{\dfrac{6+3x}x}{x + 2}$

$\displaystyle f'(-2) = \lim_{x\to-2}\frac{6+3x}{x(x + 2)}$

$\displaystyle f'(-2) = \lim_{x\to-2}\frac{3(x+2)}{x(x + 2)}$

$\displaystyle f'(-2) = \lim_{x\to-2}\frac{3}{x} = \boxed{-\dfrac32}$