(a) State the differentiation rule for the derivative of a composite function…

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

(a) State the differentiation rule for the derivative of a composite function

F(g(x))

. (a) State the differentiation rule for the derivative of a composite function

F(g(x))

. (b) Consider the integral

\displaystyle\int 3x^{2}\,e^{x^{3}}\,dx

. Identify a suitable substitution

u=\,?

so that the entire integrand, including

dx

, can be rewritten wholly in terms of

u

. Write the transformed integral in terms of

u

(evaluation not required). (c) Give a brief explanation—using part (a)—of why replacing

u

x^{3}

in the antiderivative obtained in part (b) produces a valid antiderivative of the original integrand.