Use the product Rule to find the derivative of the function $f ( x ) = x \left( x ^ { 2 } + 3 \right)$ .
A) $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 3$
B) $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 1$
C) $f ^ { \prime } ( x ) = x ^ { 2 } + 3$
D) $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3$
E) $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1$

Question

Quizplus · Accepted Answer

The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Applying this to $f(x) = x(x^2 + 3)$, we get $f'(x) = 1 \cdot (x^2 + 3) + x \cdot 2x = x^2 + 3 + 2x^2 = 3x^2 + 3$.

Use the Product Rule to Find the Derivative of the Function