Given: [tex]f(x)=\frac{3}{x}[/tex]
Determine [tex]f^{\prime}(x)[/tex] from first principles.
Answer (1)
Use 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: h x + h 3 − x 3 = x ( x + h ) − 3 .
Evaluate the limit as h approaches 0: f ′ ( x ) = − x 2 3 .
f ′ ( x ) = − x 2 3
Explanation
Problem Analysis 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. This means we will use the limit definition of the derivative.
Definition of Derivative The definition of the derivative from first principles is: f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) We will substitute the given function into this definition.
Substitution Substituting f ( x ) = x 3 into the definition, we get: f ′ ( x ) = h → 0 lim h x + h 3 − x 3 Now we need to simplify the expression inside the limit.
Simplification To simplify the expression, we first 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 divide by h :
h x ( x + h ) − 3 h = 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 So, the derivative of f ( x ) 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 estimates the cost of producing one additional item. This concept helps businesses make informed decisions about production levels.
