Given $f(x)=8-4 x^2$, find $f^{\prime}(x)$ using the limit definition of the derivative.
$f^{\prime}(x) =$
Answer (2)
Find f ( x + h ) : Substitute x + h into f ( x ) to get f ( x + h ) = 8 − 4 ( x + h ) 2 = 8 − 4 x 2 − 8 x h − 4 h 2 .
Calculate f ( x + h ) − f ( x ) : Subtract f ( x ) from f ( x + h ) to get − 8 x h − 4 h 2 .
Divide by h : Divide the result by h to get h f ( x + h ) − f ( x ) = − 8 x − 4 h .
Take the limit as h → 0 : Evaluate the limit to find f ′ ( x ) = lim h → 0 ( − 8 x − 4 h ) = − 8 x .
Therefore, f ′ ( x ) = − 8 x .
Explanation
Problem Analysis We are given the function f ( x ) = 8 − 4 x 2 and we need to find its derivative f ′ ( x ) using the limit definition of the derivative.
Limit Definition of Derivative The limit definition of the derivative is given by: f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) We will use this definition to find the derivative of the given function.
Calculate f(x+h) First, we need to find f ( x + h ) . We substitute x + h into the function: f ( x + h ) = 8 − 4 ( x + h ) 2 = 8 − 4 ( x 2 + 2 x h + h 2 ) = 8 − 4 x 2 − 8 x h − 4 h 2
Calculate f(x+h) - f(x) Next, we find the difference f ( x + h ) − f ( x ) :
f ( x + h ) − f ( x ) = ( 8 − 4 x 2 − 8 x h − 4 h 2 ) − ( 8 − 4 x 2 ) = − 8 x h − 4 h 2
Calculate [f(x+h) - f(x)] / h Now, we divide the difference by h :
h f ( x + h ) − f ( x ) = h − 8 x h − 4 h 2 = − 8 x − 4 h
Calculate the Limit Finally, we take the limit as h approaches 0: f ′ ( x ) = h → 0 lim ( − 8 x − 4 h ) = − 8 x − 4 ( 0 ) = − 8 x
Final Answer Therefore, the derivative of f ( x ) = 8 − 4 x 2 is f ′ ( x ) = − 8 x .
Examples
Understanding derivatives is crucial in many real-world applications. For example, if f ( x ) represents the position of a car at time x , then f ′ ( x ) represents the velocity of the car at time x . Knowing the velocity allows us to optimize routes, avoid collisions, and improve fuel efficiency. 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 and pricing strategies.
The derivative of the function f ( x ) = 8 − 4 x 2 using the limit definition is f ′ ( x ) = − 8 x . This was determined by substituting into the limit definition, simplifying, and taking the limit as h approaches 0.
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