Find the derivative of the function. \[ \begin{array}{l} F(\theta)=\arcsin (\sqrt{\sin (13 \theta)}) \\ F^{\prime}(\theta)= \end{array} \]

Step 1 ：Given the function \(F(\theta)=\arcsin (\sqrt{\sin (13 \theta)})\), we are asked to find its derivative \(F^{\prime}(\theta)\).

Step 2 ：To find the derivative of the function, we need to use the chain rule. The chain rule is a formula to compute the derivative of a composite function.

Step 3 ：The outer function is the arcsin function and the inner function is the square root of sin(13θ). We will first take the derivative of the outer function and then multiply it by the derivative of the inner function.

Step 4 ：The derivative of arcsin(x) is \( \frac{1}{\sqrt{1-x²}} \). The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \). The derivative of sin(x) is cos(x).

Step 5 ：So, we will first take the derivative of arcsin(√sin(13θ)) which is \( \frac{1}{\sqrt{1-(\sqrt{\sin(13θ)})²}} \). Then we will multiply it by the derivative of \( \sqrt{\sin(13θ)} \) which is \( \frac{1}{2\sqrt{\sin(13θ)}} \). Finally, we will multiply it by the derivative of sin(13θ) which is 13cos(13θ).

Step 6 ：Combining all these, we get the derivative of the function as \( \frac{13 \cos(13 \theta)}{2 \sqrt{1 - \sin(13 \theta)} \sqrt{\sin(13 \theta)}} \).

Step 7 ：\(\boxed{\frac{13 \cos(13 \theta)}{2 \sqrt{1 - \sin(13 \theta)} \sqrt{\sin(13 \theta)}}}\) is the final answer.