Find the derivative of the given function.
[tex]
\begin{array}{c}
f(x)=\left(3 x^2+7\right)(6 x+2) \\
f^{\prime}(x)=[?]\left(3 x^2+7\right)+\quad x(6 x+2)
\end{array}
[/tex]
Answer (1)
∙ Apply the product rule: f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) , where u ( x ) = 3 x 2 + 7 and v ( x ) = 6 x + 2 .
∙ Find the derivatives: u ′ ( x ) = 6 x and v ′ ( x ) = 6 .
∙ Substitute into the product rule: f ′ ( x ) = ( 6 x ) ( 6 x + 2 ) + ( 3 x 2 + 7 ) ( 6 ) .
∙ Rewrite in the required format to identify the missing term: 6 .
Explanation
Problem Analysis We are given the function f ( x ) = ( 3 x 2 + 7 ) ( 6 x + 2 ) and asked to find its derivative f ′ ( x ) and express it in the form f ′ ( x ) = [ ?] ( 3 x 2 + 7 ) + x ( 6 x + 2 ) . This problem requires us to apply the product rule of differentiation.
Applying the Product Rule The product rule states that if f ( x ) = u ( x ) v ( x ) , then f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) . In our case, let u ( x ) = 3 x 2 + 7 and v ( x ) = 6 x + 2 .
Finding the Derivatives Now we need to find the derivatives of u ( x ) and v ( x ) .
The derivative of u ( x ) = 3 x 2 + 7 is u ′ ( x ) = 6 x .
The derivative of v ( x ) = 6 x + 2 is v ′ ( x ) = 6 .
Applying the Product Rule Formula Using the product rule, we have:
f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) = ( 6 x ) ( 6 x + 2 ) + ( 3 x 2 + 7 ) ( 6 )
Finding the Missing Term We want to express f ′ ( x ) in the form f ′ ( x ) = [ ?] ( 3 x 2 + 7 ) + ( 6 x ) ( 6 x + 2 ) . Comparing this with our result f ′ ( x ) = ( 6 x ) ( 6 x + 2 ) + ( 3 x 2 + 7 ) ( 6 ) , we can see that the missing term [?] is 6.
Therefore, f ′ ( x ) = 6 ( 3 x 2 + 7 ) + 6 x ( 6 x + 2 ) .
Final Answer Thus, the derivative of the given function is f ′ ( x ) = 6 ( 3 x 2 + 7 ) + 6 x ( 6 x + 2 ) , and the missing term is 6.
So, the answer is 6 .
Examples
Understanding derivatives is crucial in physics, especially when analyzing motion. For example, if f ( x ) represents the position of an object at time x , then f ′ ( x ) gives the object's velocity at that time. By finding the derivative of a position function, we can determine how fast the object is moving at any given moment, which is essential for predicting its future location or understanding its dynamics.
