Evaluate the function at the given value.
[tex]
\begin{array}{c}
f(x)=(2 x-8)^2\left(x^3+4 x-6\right)^3 \\
f^{\prime}(1)=[?]
\end{array}
[/tex]
Answer (1)
Apply the product rule: f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) , where u ( x ) = ( 2 x − 8 ) 2 and v ( x ) = ( x 3 + 4 x − 6 ) 3 .
Find u ′ ( x ) and v ′ ( x ) using the chain rule: u ′ ( x ) = 8 x − 32 and v ′ ( x ) = 3 ( x 3 + 4 x − 6 ) 2 ( 3 x 2 + 4 ) .
Substitute x = 1 into f ′ ( x ) : f ′ ( 1 ) = ( 8 ( 1 ) − 32 ) (( 1 ) 3 + 4 ( 1 ) − 6 ) 3 + ( 2 ( 1 ) − 8 ) 2 [ 3 (( 1 ) 3 + 4 ( 1 ) − 6 ) 2 ( 3 ( 1 ) 2 + 4 )] .
Simplify to find the numerical value: f ′ ( 1 ) = 780 .
780
Explanation
Problem Analysis We are given the function f ( x ) = ( 2 x − 8 ) 2 ( x 3 + 4 x − 6 ) 3 and asked to find f ′ ( 1 ) . This requires us to find the derivative of f ( x ) and then evaluate it at x = 1 . We will use the product rule and the chain rule.
Applying the Product Rule Let u ( x ) = ( 2 x − 8 ) 2 and v ( x ) = ( x 3 + 4 x − 6 ) 3 . Then f ( x ) = u ( x ) v ( x ) , and by the product rule, f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) .
Finding u'(x) First, we find u ′ ( x ) . Using the chain rule, u ′ ( x ) = 2 ( 2 x − 8 ) ( 2 ) = 4 ( 2 x − 8 ) = 8 x − 32 .
Finding v'(x) Next, we find v ′ ( x ) . Using the chain rule, v ′ ( x ) = 3 ( x 3 + 4 x − 6 ) 2 ( 3 x 2 + 4 ) .
Substituting into the Product Rule Now we substitute u ′ ( x ) , v ( x ) , u ( x ) , and v ′ ( x ) into the product rule formula: f ′ ( x ) = ( 8 x − 32 ) ( x 3 + 4 x − 6 ) 3 + ( 2 x − 8 ) 2 [ 3 ( x 3 + 4 x − 6 ) 2 ( 3 x 2 + 4 )] .
Evaluating at x=1 We want to find f ′ ( 1 ) , so we substitute x = 1 into the expression for f ′ ( x ) : f ′ ( 1 ) = ( 8 ( 1 ) − 32 ) (( 1 ) 3 + 4 ( 1 ) − 6 ) 3 + ( 2 ( 1 ) − 8 ) 2 [ 3 (( 1 ) 3 + 4 ( 1 ) − 6 ) 2 ( 3 ( 1 ) 2 + 4 )] .
Simplifying the Expression Now we simplify the expression: f ′ ( 1 ) = ( 8 − 32 ) ( 1 + 4 − 6 ) 3 + ( 2 − 8 ) 2 [ 3 ( 1 + 4 − 6 ) 2 ( 3 + 4 )] = ( − 24 ) ( − 1 ) 3 + ( − 6 ) 2 [ 3 ( − 1 ) 2 ( 7 )] = ( − 24 ) ( − 1 ) + 36 [ 3 ( 1 ) ( 7 )] = 24 + 36 ( 21 ) = 24 + 756 = 780 .
Final Answer Therefore, f ′ ( 1 ) = 780 .
Examples
In physics, if f ( x ) represents the position of an object at time x , then f ′ ( x ) represents the velocity of the object at time x . Evaluating f ′ ( 1 ) would give the velocity of the object at time x = 1 . This is a fundamental concept in kinematics, where understanding rates of change is crucial for predicting the motion of objects. For example, if f ( x ) describes the trajectory of a rocket, f ′ ( 1 ) would tell us how fast the rocket is moving one second after launch.
