Find the maximum rate of change of the function $f(x, y)=\frac{5 y^{2}}{x}$ at the point $(4,2)$

Step 1 ：Given the function \(f(x, y)=\frac{5 y^{2}}{x}\), we are asked to find the maximum rate of change at the point (4,2).

Step 2 ：The maximum rate of change of a function at a given point is given by the gradient of the function at that point. The gradient is a vector that points in the direction of the steepest ascent of the function, and its magnitude is the maximum rate of change.

Step 3 ：To find the gradient, we need to compute the partial derivatives of the function with respect to each variable. In this case, we need to compute the partial derivatives of \(f(x, y)\) with respect to \(x\) and \(y\).

Step 4 ：The partial derivative of \(f(x, y)\) with respect to \(x\) is \(-\frac{5y^{2}}{x^{2}}\) and with respect to \(y\) is \(\frac{10y}{x}\).

Step 5 ：Substituting the point (4,2) into the partial derivatives, we get \(-\frac{5}{4}\) and 5 respectively.

Step 6 ：The gradient at the point (4,2) is then the vector \(-\frac{5}{4}, 5\).

Step 7 ：The magnitude of the gradient vector at this point is given by the square root of the sum of the squares of these partial derivatives. This will give us the maximum rate of change of the function at the point (4,2).

Step 8 ：Calculating the magnitude, we get approximately 5.153882032022076.

Step 9 ：Final Answer: The maximum rate of change of the function \(f(x, y)=\frac{5 y^{2}}{x}\) at the point \((4,2)\) is \(\boxed{5.153882032022076}\).