Answer
In both cases (a. and b.):
$f^{\prime}(x)=3x^{2}$
Work Step by Step
a.
Since $ x\cdot x^{2}=x^{3}$, we use the Power Rule:
$g(x)=x^{3}$
$g^{\prime}(x)=3\cdot x^{2}$
b.
$u(x)=x ,\ \ \ v(x)=x^{2}$
$f(x)=x\cdot x^{2}=u(x)v(x)$
$u^{\prime}(x)=1,\ \ \ v^{\prime}(x)=2x$
Product Rule:
$f^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)$
$= 1(x^{2})+x(2x)=x^{2}+2x^{2}=3x^{2}$
