Which of the following best explains why the derivative of does not exist at ?
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Which of the following best explains why the derivative of f(x)=∣x∣ does not exist at x=0?
(a) The tangent line at x=0 is vertical, so the slope is infinite on both sides of the point.
(b) The function has a jump discontinuity at x=0, therefore it cannot be differentiable there.
(c) The left-hand derivative equals −1 and the right-hand derivative equals 1, so the two one–sided derivatives are unequal.
(d) The limit in the derivative definition produces the indeterminate form 0/0 at x=0, so a numeric derivative cannot be found.