Sure! Let’s go through the process of differentiating and then integrating a simple function step by step. We’ll use the function ( f(x) = x^2 ) as our example.

1. Differentiation (Finding the Derivative)

Forward Approach:

To find the derivative ( f’(x) ), which represents the rate of change of ( f(x) ) with respect to ( x ):

1. Start with the function ( f(x) ): [ f(x) = x^2 ]

2. Apply the Power Rule for Differentiation:

   • For a function of the form ( f(x) = x^n ), the derivative is ( f’(x) = n x^{n-1} ).
   • Apply this rule to ( f(x) = x^2 ): [ f’(x) = 2x ]

3. Interpretation:

   • ( f’(x) = 2x ) tells us that the slope (rate of change) of the function ( f(x) = x^2 ) at any point ( x ) is ( 2x ).

Backward Approach:

To verify the result by integrating ( f’(x) ) back to ( f(x) ):

1. Start with ( f’(x) = 2x ):

2. Integrate ( f’(x) ) to find ( f(x) ):

   • To find ( f(x) ), integrate ( f’(x) = 2x ) with respect to ( x ): [ f(x) = 2x , dx ]

3. Perform the Integration:

   • Integrate term by term: [ f(x) = 2 x , dx ]
   • Apply the power rule of integration: ( x^n , dx = + C ), where ( C ) is the constant of integration.
   • For ( n = 1 ): [ x , dx = + C ]
   • Therefore, [ f(x) = 2 ( ) + C = x^2 + C ]

4. Determine the Constant of Integration ( C ):

   • The constant ( C ) is arbitrary and represents the family of functions that have the same derivative ( 2x ).
   • ( C ) can be determined if initial conditions (e.g., ( f(0) = 0 )) are provided, but in general, it remains unspecified unless specified by additional information.

Summary:

• Differentiation (Forward): ( f’(x) = 2x )
• Integration (Backward): ( f(x) = x^2 + C )

These steps demonstrate the process of differentiating and integrating a simple function ( f(x) = x^2 ) using both forward (differentiation) and backward (integration) approaches. This process is foundational in calculus and essential for understanding how functions change and accumulate over their domains.