If [tex]f(x)=14[/tex], find [tex]f^{\prime}(-7)[/tex]
Answer (1)
Recognize that f ( x ) = 14 is a constant function.
Recall that the derivative of a constant function is zero: f ′ ( x ) = 0 .
Evaluate the derivative at x = − 7 : f ′ ( − 7 ) = 0 .
State the final answer: 0 .
Explanation
Understanding the Problem We are given the function f ( x ) = 14 . This is a constant function, meaning that its value is the same regardless of the input x . We are asked to find the derivative of this function evaluated at x = − 7 , which is denoted as f ′ ( − 7 ) .
Finding the Derivative The derivative of a constant function is always zero. This is because the rate of change of a constant function is zero; the function's value never changes as x changes. Therefore, f ′ ( x ) = 0 for all x .
Evaluating the Derivative at x = -7 Since the derivative f ′ ( x ) is 0 for all x , it follows that f ′ ( − 7 ) = 0 . No matter what value we plug in for x , the derivative will always be 0.
Final Answer Therefore, the derivative of the function f ( x ) = 14 evaluated at x = − 7 is 0.
Examples
Imagine you are tracking the temperature in a room, and it remains constant at 14 degrees Celsius throughout the day. The rate of change of the temperature is zero, because the temperature is not changing. This is analogous to the derivative of a constant function being zero. Understanding this concept is crucial in many real-world applications, such as physics, engineering, and economics, where constant values and their rates of change are frequently encountered.
