A student claims that an antiderivative of is

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A student claims that an antiderivative of

4x^{3}\sqrt{x^{4}+1}

\displaystyle \frac{2}{3}(x^{4}+1)^{3/2}+C

. Which of the following statements correctly verifies the student's claim? (a) The claim is incorrect since

\displaystyle \frac{d}{dx}\!\left[\frac{2}{3}(x^{4}+1)^{3/2}\right]=6x^{3}\sqrt{x^{4}+1}

, not

4x^{3}\sqrt{x^{4}+1}

. (b) The claim is correct because

\displaystyle \frac{d}{dx}\!\left[\frac{2}{3}(x^{4}+1)^{3/2}\right]=4x^{3}\sqrt{x^{4}+1}

. (c) The claim is incorrect because differentiating produces only

\sqrt{x^{4}+1}

; the

4x^{3}

factor from the Chain Rule is missing. (d) The claim is incorrect since

\displaystyle \frac{d}{dx}\!\left[\frac{2}{3}(x^{4}+1)^{3/2}\right]=4x^{3}(x^{4}+1)^{3/2}

, so the exponent remains

\tfrac{3}{2}

, not

\tfrac{1}{2}