The slope of the graph of the function \( g(x)=\frac{7 x}{x+4} \) at \( (3,3) \) is (Type an integer or a simplified fraction.)

To find the slope of the graph of the function \( g(x) = \frac{7x}{x+4} \) at the point \( (3, 3) \), we need to calculate the derivative \( g'(x) \). We can use the quotient rule, which states: \[ g'(x) = \frac{(f'(x)g(x) - f(x)g'(x))}{(g(x))^2} \] where \( f(x) = 7x \) and \( g(x) = x + 4 \). Thus, \( f'(x) = 7 \) and \( g'(x) = 1 \). Applying the quotient rule, we get: \[ g'(x) = \frac{(7)(x + 4) - (7x)(1)}{(x + 4)^2} = \frac{7x + 28 - 7x}{(x + 4)^2} = \frac{28}{(x + 4)^2} \] Next, we evaluate this derivative at \( x = 3 \): \[ g'(3) = \frac{28}{(3 + 4)^2} = \frac{28}{7^2} = \frac{28}{49} = \frac{4}{7} \] Thus, the slope of the graph at the point \( (3, 3) \) is \( \frac{4}{7} \).