\( \Gamma ( - \frac { 2 } { 3 } ) \frac { \neq ( x + h ) - f ( x ) } { h } \)

The problem seems to involve the gamma function and possibly a derivative. However, the expression on the left-hand side, \( \Gamma(-\frac{2}{3}) \), is not equal to the expression on the right-hand side, \( \frac{(x+h) - f(x)}{h} \). These are two completely different mathematical objects. The gamma function, \( \Gamma(z) \), is a special function defined for complex numbers \( z \) with a real part greater than 0. It is not a function of \( x \) or \( h \), and it is not related to the difference quotient, which is a way to approximate the derivative of a function. If you are trying to show that \( \Gamma(-\frac{2}{3}) \) is not equal to the difference quotient, you can simply state that they are different functions and provide their definitions or properties to support your claim. Here is a brief explanation of each: 1. The gamma function, \( \Gamma(z) \), is defined as: \[ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt \] for \( \Re(z) > 0 \). 2. The difference quotient is a way to approximate the derivative of a function \( f(x) \) at a point \( x \) using the limit as \( h \) approaches 0: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Since \( \Gamma(-\frac{2}{3}) \) is a specific value of the gamma function and \( \frac{(x+h) - f(x)}{h} \) is a function of \( x \) and \( h \), they are not equal.