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)(f(x) - f(-2))/(x - (-2))`

`\displaystyle f'(-2) = \lim_(x\to-2)(\frac6x + 3)/(x + 2)`

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

`\displaystyle f'(-2) = \lim_(x\to-2)(6+3x)/(x(x + 2))`

`\displaystyle f'(-2) = \lim_(x\to-2)(3(x+2))/(x(x + 2))`

`\displaystyle f'(-2) = \lim_(x\to-2)(3)/(x) = \boxed{-\frac32}`