The strength of a patient's reaction to a dose of $x$ milligrams of a certain drug is $R(x)=4 x \sqrt{11+0.5 x}$ for $0 \leq x \leq 180$. The derivative $R^{\prime}(x)$ is called the sensitivity to the drug. Find $R^{\prime}(50)$, the sensitivity to a dose of $50 \mathrm{mg}$.
\[
R^{\prime}(50)=
\]
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Step 1 ：We are given the function \(R(x)=4 x \sqrt{11+0.5 x}\) which represents the strength of a patient's reaction to a dose of a certain drug. We are asked to find the sensitivity to a dose of 50mg, which is represented by the derivative of the function at \(x=50\), denoted as \(R^{\prime}(50)\).

Step 2 ：To find \(R^{\prime}(50)\), we first need to find the derivative of the function \(R(x)\). This can be done using the rules of differentiation, specifically the product rule and the chain rule.

Step 3 ：The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 4 ：Applying these rules, we find that the derivative of \(R(x)\) is \(R^{\prime}(x) = 0.0909090909090909\sqrt{11}x/\sqrt{0.0454545454545455x + 1} + 4\sqrt{11}\sqrt{0.0454545454545455x + 1}\).

Step 5 ：We then substitute \(x=50\) into the derivative to find \(R^{\prime}(50)\).

Step 6 ：Doing this, we find that \(R^{\prime}(50) = 9.74886680801436\sqrt{11}\).

Step 7 ：Thus, the sensitivity to a dose of 50mg is approximately \(\boxed{9.74886680801436 \sqrt{11}}\).