Differentiate [tex]F^{\prime}(x)=x^3[/tex] from first principle

The derivative of F ′ ( x ) = x 3 from first principles is calculated using the limit definition of a derivative, leading to the final result of 3 x 2 .
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Answered by Anonymous | 2025-07-08

Apply the first principle definition of a derivative: f ′ ( x ) = lim h → 0 ​ h f ( x + h ) − f ( x ) ​ .
Substitute f ( x ) = x 3 into the definition and expand ( x + h ) 3 .
Simplify the expression by factoring out and canceling h .
Evaluate the limit as h approaches 0 to find the derivative: 3 x 2 ​ .

Explanation

Problem Analysis We are asked to find the derivative of F ′ ( x ) = x 3 using the first principle definition of a derivative. This means we need to apply the limit definition of the derivative.

First Principle Definition The definition of the derivative from first principles is: f ′ ( x ) = h → 0 lim ​ h f ( x + h ) − f ( x ) ​ In this case, f ( x ) = F ′ ( x ) = x 3 .

Substitution and Expansion Substitute f ( x ) = x 3 into the definition: f ′ ( x ) = h → 0 lim ​ h ( x + h ) 3 − x 3 ​ Expand ( x + h ) 3 : ( x + h ) 3 = x 3 + 3 x 2 h + 3 x h 2 + h 3

Simplification Substitute the expanded form back into the limit: f ′ ( x ) = h → 0 lim ​ h x 3 + 3 x 2 h + 3 x h 2 + h 3 − x 3 ​ Simplify the expression: f ′ ( x ) = h → 0 lim ​ h 3 x 2 h + 3 x h 2 + h 3 ​ Factor out h from the numerator: f ′ ( x ) = h → 0 lim ​ h h ( 3 x 2 + 3 x h + h 2 ) ​ Cancel out h : f ′ ( x ) = h → 0 lim ​ ( 3 x 2 + 3 x h + h 2 )

Evaluation of the Limit Evaluate the limit as h approaches 0: f ′ ( x ) = 3 x 2 + 3 x ( 0 ) + ( 0 ) 2 = 3 x 2 Therefore, the derivative of F ′ ( x ) = x 3 is 3 x 2 .

Final Answer The derivative of F ′ ( x ) = x 3 from first principles is: 3 x 2 ​


Examples
In physics, if x 3 represents the distance an object travels as a function of time, then its derivative, 3 x 2 , represents the object's velocity at any given time. Understanding derivatives from first principles helps in modeling motion and understanding rates of change in various physical systems, such as calculating the speed of a car or the growth rate of a population.

Answered by GinnyAnswer | 2025-07-08