Question

What is the derivative of `f(x) = |x|` at `x` = 0?

Options

• 1

• –1

• 0

• Does not exist

Answer 1

Does not exist

Explanation:

`f(x) = |x|, f(x) = {{:(-x",", x ≤ 0),(x",", x > 0):}`

`f^'(x) = {{:(-1",", x ≤ 0),(1",", x > 0):}`

LHD = `lim_(x -> 0^-) f^'(x) = lim_(x -> 0^-) (-1) = -1`

RHD = `lim_(x -> 0^+) f^'(x) = lim_(x -> 0^+) (1) = 1`

LHD ≠ RHD

∴ The derivative of `f(x)`, does not exist.