Find the critical points of the following function. \[ f(x)=\frac{8 x}{x^{2}+64} \] What is the derivative of $f(x)=\frac{8 x}{x^{2}+64} ?$ \[ f^{\prime}(x)=\square \]

Step 1 ：We are given the function \(f(x)=\frac{8 x}{x^{2}+64}\). We need to find its critical points. To do this, we first need to find its derivative.

Step 2 ：The derivative of a function at a certain point gives the slope of the tangent line at that point. The critical points of a function are the points where the derivative is zero or undefined.

Step 3 ：To find the derivative of the function, we will use the quotient rule for differentiation. The quotient rule states that the derivative of \(\frac{u}{v}\) is \(\frac{u'v - uv'}{v^2}\), where \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives.

Step 4 ：In our function, \(u = 8x\) and \(v = x^2 + 64\). The derivative of \(u\) with respect to \(x\) is \(8\), and the derivative of \(v\) with respect to \(x\) is \(2x\).

Step 5 ：Substituting these values into the quotient rule, we get \(f'(x) = \frac{8*(x^2 + 64) - 8x*2x}{(x^2 + 64)^2}\).

Step 6 ：Simplifying this expression, we find that the derivative of the function \(f(x)=\frac{8 x}{x^{2}+64}\) is \(f'(x) = \frac{512 - 8x^2}{(x^2 + 64)^2}\).

Step 7 ：\(\boxed{f'(x) = \frac{512 - 8x^2}{(x^2 + 64)^2}}\) is the final answer.