Find the slope of the graph of the function \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \). Then find an equation for the line tangent The slope of the graph of the function \( g(x)=\frac{3 x}{x-2} \) at \( (4,6) \) is (Type an integer or a simplified fraction.)

To find the slope of the graph of the function \( g(x) = \frac{3x}{x-2} \) at the point \( (4,6) \), we first need to compute the derivative, \( g'(x) \). Using the quotient rule, which states \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \), where \( u = 3x \) and \( v = x-2 \): 1. Compute \( u' = 3 \) and \( v' = 1 \). 2. Apply the quotient rule: \[ g'(x) = \frac{3(x-2) - 3x(1)}{(x-2)^2} = \frac{3x - 6 - 3x}{(x-2)^2} = \frac{-6}{(x-2)^2} \] Now substitute \( x = 4 \) into \( g'(x) \): \[ g'(4) = \frac{-6}{(4-2)^2} = \frac{-6}{2^2} = \frac{-6}{4} = -\frac{3}{2} \] So, the slope of the graph at \( (4, 6) \) is \( -\frac{3}{2} \). Next, we find the equation of the tangent line at this point using the point-slope form \( y - y_1 = m(x - x_1) \): \[ y - 6 = -\frac{3}{2}(x - 4) \] Distributing the slope: \[ y - 6 = -\frac{3}{2}x + 6 \] Adding 6 to both sides: \[ y = -\frac{3}{2}x + 12 \] Thus, the slope of the graph of the function \( g(x) = \frac{3x}{x-2} \) at \( (4,6) \) is \( -\frac{3}{2} \), and the equation of the tangent line is \( y = -\frac{3}{2}x + 12 \).