For the function

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

For the function f(x)=x3f(x)=\sqrt[3]{x}, which of the following best describes f′(0)f'(0)? (a) f′(0)=0f'(0)=0 because the tangent line is horizontal at the origin. (b) f′(0)=∞f'(0)=\infty since f′(x)=13x2/3\displaystyle f'(x)=\dfrac{1}{3x^{2/3}} and substituting x=0x=0 gives division by zero. (c) f′(0)f'(0) does not exist because the slope of the secant lines approaches ±∞\pm\infty as x→0x\to 0. (d) f′(0)f'(0) does not exist because ff is discontinuous at x=0x=0.

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