Suppose a classmate claims that an antiderivative of the function f(x)= 3x^ 2 (x^ 3…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Suppose a classmate claims that an antiderivative of the function

f(x)=\frac{3x^{2}}{(x^{3}+1)^{5/2}}

F(x)=-(x^{3}+1)^{-3/2}+C,

where

C

is an arbitrary constant. (1) Differentiate

F(x)

explicitly and simplify the result. (2) Compare your derivative to

f(x)

. State whether the classmate’s claim is correct and justify your conclusion in complete sentences.