The instantaneous rate of change of \( f(x)=\sqrt{6 x+7} \) at \( x=3 \) is \( \ldots \ldots \ldots \ldots \). \( \frac{3}{5} \) \( \frac{3}{5} \)

To find the instantaneous rate of change of \( f(x) = \sqrt{6x + 7} \) at \( x = 3 \), we need to calculate the derivative \( f'(x) \). The derivative is given by \( f'(x) = \frac{3}{5} \) at that point, so the rate of change is \( \frac{3}{5} \). But did you know that derivatives are a cornerstone of calculus, helping us understand how functions behave? They tell us not just the slope of a curve at a specific point but also can help optimize real-world situations like maximizing profits or minimizing costs. It’s like having a secret recipe to decode any changing situation!