If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$.
Select the correct answer.
A) $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 2 } f ( x ) + x ^ { 3 } f ^ { \prime } ( x )$
B) $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) + x ^ { 2 } f ^ { \prime } ( x )$
C) $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 2 x ^ { 2 } f ( x ) - x ^ { 3 } f ^ { \prime } ( x )$
D) $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 2 } f ( x ) - x ^ { 3 } f ^ { \prime } ( x )$
E) $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) - x ^ { 2 } f ^ { \prime } ( x )$

Question

Quizplus · Accepted Answer

The product rule states that the derivative of $y = u(x)v(x)$ is $\frac { d } { d x } \left( u(x)v(x) \right) = u(x)v'(x) + v(x)u'(x)$. In this case, $u(x) = x^3$ and $v(x) = f(x)$, so $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = x^3f'(x) + f(x)3x^2 = 3x^2f(x) + x^3f'(x)$.

If fff Is a Differentiable Function, Find an Expression for the Derivative

If $f$ Is a Differentiable Function, Find an Expression for the Derivative