Given: [tex]f(x)=\frac{3}{x}[/tex]
Determine [tex]f^{\prime}(x)[/tex] from first principles.
Answer (1)
Apply the definition of the derivative: f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) .
Substitute f ( x ) = x 3 into the definition: f ′ ( x ) = lim h → 0 h x + h 3 − x 3 .
Simplify the expression: f ′ ( x ) = lim h → 0 x ( x + h ) − 3 .
Evaluate the limit as h approaches 0: f ′ ( x ) = x 2 − 3 .
f ′ ( x ) = x 2 − 3
Explanation
Understanding the Problem and the Definition of the Derivative We are given the function f ( x ) = x 3 and we want 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 ) This formula calculates the instantaneous rate of change of the function at a specific point by considering the limit of the average rate of change as the interval approaches zero.
Substituting the Function into the Definition Now, let's substitute f ( x ) = x 3 into the definition of the derivative: f ′ ( x ) = h → 0 lim h x + h 3 − x 3 We need to simplify the expression inside the limit to evaluate it.
Simplifying the Expression To simplify the expression, we first find a common denominator for the fractions in the numerator: f ′ ( x ) = h → 0 lim h x ( x + h ) 3 x − 3 ( x + h ) Now, we simplify the numerator: f ′ ( x ) = h → 0 lim h x ( x + h ) 3 x − 3 x − 3 h = h → 0 lim h x ( x + h ) − 3 h
Dividing by h and Cancelling Terms Next, we divide by h , which is the same as multiplying by h 1 :
f ′ ( x ) = h → 0 lim h ⋅ x ( x + h ) − 3 h Now, we can cancel h from the numerator and the denominator: f ′ ( x ) = h → 0 lim x ( x + h ) − 3
Evaluating the Limit and Finding the Derivative Finally, we evaluate the limit as h approaches 0: f ′ ( x ) = x ( x + 0 ) − 3 = x 2 − 3 So, the derivative of f ( x ) = x 3 is f ′ ( x ) = x 2 − 3 .
Final Answer Therefore, the derivative of the function f ( x ) = x 3 from first principles is: f ′ ( x ) = x 2 − 3
Examples
Understanding derivatives is crucial in many real-world applications. For instance, if f ( x ) represents the position of a car at time x , then f ′ ( x ) gives the car's velocity at time x . In economics, if f ( x ) is the cost of producing x items, then f ′ ( x ) is the marginal cost, which tells us approximately how much it costs to produce one more item. Derivatives are also used extensively in physics, engineering, and computer science to model and optimize various systems.
