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−1 does not exist at x=0?
(a) Applying the power rule gives f′(x)=−x−2; plugging in x=0 gives 0, so the derivative exists and equals 0.
(b) The difference quotient cannot be formed because f(0) is undefined, so the limit that defines f′(0) does not exist.
(c) As x approaches 0, the slopes become arbitrarily large in magnitude, so the derivative is infinite, not nonexistent.
(d) Because the denominator (x−0) in the difference quotient equals 0 at x=0, the quotient itself is undefined and a derivative cannot be computed.