The tangent line to the graph of function \( g \) at the point \( (1,5) \) passes through.the point \( (3,4) \). Find \( g^{\prime}(1) \). \( g^{\prime}(1)= \)

To find \( g^{\prime}(1) \), we need to determine the slope of the tangent line to the graph of the function \( g \) at the point \( (1, 5) \). The slope of the tangent line is given by the derivative \( g^{\prime}(1) \). We know that the tangent line passes through the points \( (1, 5) \) and \( (3, 4) \). We can calculate the slope of the line that connects these two points using the formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in our points \( (1, 5) \) and \( (3, 4) \): \[ \text{slope} = \frac{4 - 5}{3 - 1} = \frac{-1}{2} \] Since the slope of the tangent line at the point \( (1, 5) \) is equal to the slope we just calculated, we have: \[ g^{\prime}(1) = -\frac{1}{2} \] Thus, the value of \( g^{\prime}(1) \) is \[ \boxed{-\frac{1}{2}} \]