\( f ^ { \prime } ( x ) = \frac { f ( x + 6 ) - f ( x - 6 ) } { 2 } \)

The equation you've provided is a form of a finite difference approximation for the derivative of a function \( f(x) \). Specifically, it resembles the central difference formula, which is used to approximate the derivative of a function at a point. The equation can be interpreted as follows: \[ f'(x) = \frac{f(x + 6) - f(x - 6)}{2} \] This suggests that the derivative \( f'(x) \) is being approximated by taking the average rate of change of the function \( f \) over an interval of length 12 (from \( x - 6 \) to \( x + 6 \)). To analyze this further, we can rearrange the equation to express it in terms of the function values: \[ f(x + 6) - f(x - 6) = 2f'(x) \] This indicates that the difference in function values at points \( x + 6 \) and \( x - 6 \) is twice the derivative at point \( x \). ### Implications 1. **Approximation of Derivative**: This formula provides a way to compute the derivative of \( f \) at point \( x \) using function values at points \( x + 6 \) and \( x - 6 \). 2. **Function Behavior**: If \( f \) is a smooth function, as \( h \) (in this case, 6) approaches 0, the central difference approximation becomes more accurate. 3. **Numerical Methods**: This type of finite difference method is commonly used in numerical analysis to approximate derivatives when the function is known only at discrete points. ### Example If you have a specific function \( f(x) \), you can use this formula to compute \( f'(x) \) at any point \( x \) by evaluating \( f(x + 6) \) and \( f(x - 6) \). If you have any specific function or further questions regarding this equation, feel free to ask!