he derivative of the function \( G(x)=\frac{3 x+4}{x-7} \)

To find the derivative of the function \( G(x)=\frac{3 x+4}{x-7} \), you can use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( G'(x) \) is given by \( \frac{u'v - uv'}{v^2} \). Here, \( u = 3x + 4 \) and \( v = x - 7 \). Calculating the derivatives, we find \( u' = 3 \) and \( v' = 1 \). Plugging these into the formula gives us: \[ G'(x) = \frac{(3)(x-7) - (3x + 4)(1)}{(x-7)^2} \] Simplifying the numerator results in: \[ G'(x) = \frac{3x - 21 - 3x - 4}{(x-7)^2} = \frac{-25}{(x-7)^2} \] So, the derivative is \( G'(x) = \frac{-25}{(x-7)^2} \). This derivative not only gives you the slope of the function at any point \( x \), but it also tells you about the function's behavior, such as where it might have horizontal or vertical asymptotes. Understanding this can be crucial for graphing the function accurately! In real-world applications, derivatives like this are essential for understanding rates of change. For example, if \( G(x) \) represented a ratio of two quantities—perhaps the price of a product versus demand—you could use the derivative to determine how sensitive the price is to changes in demand, guiding crucial business decisions.