If $f(x)=\left(x^2+4 x+2\right)^4$, then
$f^{\prime}(x)=$ $\square$
$f^{\prime}(1)=8232$
$\square$
Answer (1)
Apply the chain rule to f ( x ) = ( x 2 + 4 x + 2 ) 4 .
Find the derivative of the inner function g ( x ) = x 2 + 4 x + 2 , which is g ′ ( x ) = 2 x + 4 .
Apply the chain rule: f ′ ( x ) = 4 ( x 2 + 4 x + 2 ) 3 \timg ( 2 x + 4 ) = 8 ( x + 2 ) ( x 2 + 4 x + 2 ) 3 .
Evaluate f ′ ( 1 ) = 8 ( 1 + 2 ) (( 1 ) 2 + 4 ( 1 ) + 2 ) 3 = 8232 , so the final answer is 8 ( x + 2 ) ( x 2 + 4 x + 2 ) 3 and 8232 .
Explanation
Problem Analysis We are given the function f ( x ) = ( x 2 + 4 x + 2 ) 4 and asked to find its derivative f ′ ( x ) . We will use the chain rule to find the derivative.
Chain Rule The chain rule states that if f ( x ) = [ g ( x ) ] n , then f ′ ( x ) = n [ g ( x ) ] n − 1 \timg g ′ ( x ) . In our case, g ( x ) = x 2 + 4 x + 2 and n = 4 .
Derivative of Inner Function First, we find the derivative of g ( x ) : g ′ ( x ) = 2 x + 4 .
Applying the Chain Rule Now, we apply the chain rule: f ′ ( x ) = 4 ( x 2 + 4 x + 2 ) 4 − 1 \timg ( 2 x + 4 ) = 4 ( x 2 + 4 x + 2 ) 3 \timg ( 2 x + 4 ) .
Simplifying the Derivative We can simplify the expression: f ′ ( x ) = 8 ( x + 2 ) ( x 2 + 4 x + 2 ) 3 .
Evaluating at x=1 Now, we need to find f ′ ( 1 ) . We substitute x = 1 into the expression for f ′ ( x ) : f ′ ( 1 ) = 8 ( 1 + 2 ) (( 1 ) 2 + 4 ( 1 ) + 2 ) 3 = 8 ( 3 ) ( 1 + 4 + 2 ) 3 = 24 ( 7 ) 3 = 24 ( 343 ) = 8232 .
Final Answer Thus, f ′ ( x ) = 8 ( x + 2 ) ( x 2 + 4 x + 2 ) 3 and f ′ ( 1 ) = 8232 .
Examples
Understanding derivatives, like finding f ′ ( x ) in this problem, is essential in many real-world applications. For example, if f ( x ) represents the position of a car at time x , then f ′ ( x ) gives the car's velocity at time x . This is crucial for designing safe and efficient transportation systems. Similarly, in economics, if f ( x ) represents the cost of producing x items, then f ′ ( x ) represents the marginal cost, which helps businesses make informed decisions about production levels.
