If the function given is \( f(x)=\frac{5 x+7}{4 x+2} \), then its rate of change at \( x=2 \) is ......... \( \frac{-9}{2} \) \( \frac{9}{50} \)

To find the rate of change of the function \( f(x)=\frac{5x+7}{4x+2} \) at \( x=2 \), we first need to calculate the derivative \( f'(x) \). Using the quotient rule, we have: \[ f'(x) = \frac{(4x+2)(5) - (5x+7)(4)}{(4x+2)^2} \] Now, substituting \( x=2 \) into the derivative will give us the rate of change. After simplifying the computations, we find that: \[ f'(2) = \frac{-9}{2} \] So the rate of change at \( x=2 \) is indeed \( \frac{-9}{2} \). For those looking to deepen their understanding, exploring the concept of derivatives in calculus can be enlightening! It's all about how fast a function is changing at any point. This has applications in physics, economics, and many fields where rates of change are crucial. Enjoy the journey through the math jungle! If you're keen on further studies, consider delving into real-world applications like optimization problems. They show how derivatives inform decisions in business, engineering, and even sports strategies! From maximizing profits to minimizing costs, the world of derivatives can help guide critical choices. Happy learning!