Given: [tex]f(x)=\frac{3}{x}[/tex]
Determine [tex]f^{\prime}(x)[/tex] from first principles.
Answer (1)
Find f ( x + h ) : Since f ( x ) = x 3 , then f ( x + h ) = x + h 3 .
Substitute into the definition of the derivative: f ′ ( x ) = lim h → 0 h x + h 3 − x 3 .
Simplify the expression: h x + h 3 − x 3 = x ( x + h ) − 3 .
Evaluate the limit as h approaches 0: f ′ ( x ) = lim h → 0 x ( x + h ) − 3 = − x 2 3 .
f ′ ( x ) = − x 2 3
Explanation
Understanding the Problem We are given the function f ( x ) = x 3 and we need to find its derivative f ′ ( x ) using the definition of the derivative from first principles, which is given by: f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x )
Finding f(x+h) First, we need to find f ( x + h ) . Since f ( x ) = x 3 , we have: f ( x + h ) = x + h 3 Now, we substitute f ( x + h ) and f ( x ) into the definition of the derivative: f ′ ( x ) = h → 0 lim h x + h 3 − x 3
Simplifying the Expression Next, we simplify the expression inside the limit. To do this, we find a common denominator for the fractions in the numerator: h x + h 3 − x 3 = h x ( x + h ) 3 x − 3 ( x + h ) = h x ( x + h ) 3 x − 3 x − 3 h = h x ( x + h ) − 3 h Now, we can cancel the h in the numerator and denominator: h x ( x + h ) − 3 h = x ( x + h ) − 3
Evaluating the Limit Now, we evaluate the limit as h approaches 0: f ′ ( x ) = h → 0 lim x ( x + h ) − 3 = x ( x + 0 ) − 3 = x 2 − 3 Therefore, the derivative of f ( x ) = x 3 is: f ′ ( x ) = − x 2 3
Final Answer Thus, the derivative of the function f ( x ) = x 3 from first principles is: f ′ ( x ) = − x 2 3
Examples
In physics, if f ( x ) represents the displacement of an object as a function of time x , then f ′ ( x ) represents the velocity of the object at time x . For example, if the displacement is given by f ( x ) = x 3 , then the velocity is f ′ ( x ) = − x 2 3 . This tells us how the velocity changes with time. Understanding derivatives helps in analyzing motion and other dynamic processes.
