If $f(x)=-x+24$, find $f^{\prime}(x)$ from first principles.
Answer (1)
Apply the first principle of differentiation: f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) .
Substitute f ( x ) = − x + 24 into the formula: f ′ ( x ) = lim h → 0 h − ( x + h ) + 24 − ( − x + 24 ) .
Simplify the expression: f ′ ( x ) = lim h → 0 h − h .
Evaluate the limit to find the derivative: − 1 .
Explanation
Understanding the Problem and the First Principle We are given the function f ( x ) = − x + 24 and we need to find its derivative f ′ ( x ) using the first principle of differentiation. The first principle, also known as the definition of the derivative, states that: f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) This formula calculates the instantaneous rate of change of the function f ( x ) at a particular point x .
Substituting the Function Now, let's substitute the given function f ( x ) = − x + 24 into the definition of the derivative: f ′ ( x ) = h → 0 lim h ( − ( x + h ) + 24 ) − ( − x + 24 ) This step involves replacing f ( x ) with its actual expression in the limit formula.
Simplifying the Expression Next, we simplify the expression inside the limit: f ′ ( x ) = h → 0 lim h − x − h + 24 + x − 24 Here, we expand the terms in the numerator and prepare to cancel out terms.
More Simplification Further simplification yields: f ′ ( x ) = h → 0 lim h − h Notice that − x and + x cancel out, and + 24 and − 24 also cancel out, leaving only − h in the numerator.
Evaluating the Limit Now, we can evaluate the limit: f ′ ( x ) = h → 0 lim − 1 = − 1 Since h cancels out from the numerator and denominator, we are left with − 1 . The limit of a constant is just the constant itself.
Final Result Therefore, the derivative of f ( x ) = − x + 24 is: f ′ ( x ) = − 1 This means that the rate of change of the function f ( x ) is constant and equal to − 1 .
Examples
In physics, if f ( x ) represents the position of an object at time x , then f ′ ( x ) represents the velocity of the object. In this case, if f ( x ) = − x + 24 describes the position, then f ′ ( x ) = − 1 means the object is moving at a constant velocity of -1 unit per unit time. This concept is crucial in understanding motion and rates of change in various real-world scenarios, such as calculating the speed of a car or the rate of cooling of an object.
