A student differentiates (with ) and reports

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

Problem

A student differentiates q(x)=exln⁡xq(x)=\dfrac{e^{x}}{\ln x} (with x>1x>1) and reports q′(x)=xexq'(x)=xe^{x}. Which statement best explains the student’s primary error? (a) The student differentiated only the numerator, treating ln⁡x\ln x as a constant and writing q′(x)=(ex)′ln⁡xq'(x)=\dfrac{(e^{x})'}{\ln x}. (b) The student divided the derivatives of the numerator and denominator, writing q′(x)=(ex)′(ln⁡x)′q'(x)=\dfrac{(e^{x})'}{(\ln x)'} instead of applying the quotient rule. (c) The student applied the quotient rule but forgot to square the denominator, so the (ln⁡x)2(\ln x)^2 term is missing. (d) The student replaced (ln⁡x)′(\ln x)' with xx instead of the correct 1x\dfrac{1}{x}, giving an extra factor of xx.

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