If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$.
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

If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$.
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)
$\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) - x ^ { 2 } f ^ { \prime } ( x )$

Quizplus · Accepted Answer

The product rule states that the derivative of $f(x)g(x)$ is $f'(x)g(x) + f(x)g'(x)$. In this case, $f(x) = x^3$ and $g(x) = f(x)$, so the derivative of $y = x^3 f(x)$ is $3x^2 f(x) + x^3 f'(x)$.

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

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