Find the derivative of the given function. \[ q=\sin \left(\frac{t}{\sqrt{t+2}}\right) \]

Step 1 ：We are given the function \(q=\sin \left(\frac{t}{\sqrt{t+2}}\right)\) and we are asked to find its derivative.

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. The outer function is the sine function and the inner function is \(\frac{t}{\sqrt{t+2}}\).

Step 3 ：The derivative of the sine function is the cosine function. The derivative of \(\frac{t}{\sqrt{t+2}}\) can be found using the quotient rule. The quotient rule states that the derivative of \(\frac{f}{g}\) is \(\frac{f'g - fg'}{g^2}\), where \(f'\) and \(g'\) are the derivatives of \(f\) and \(g\) respectively.

Step 4 ：Applying the chain rule, we get the derivative of the function \(q=\sin \left(\frac{t}{\sqrt{t+2}}\right)\) as \((-t/(2*(t + 2)^{3/2}) + 1/\sqrt{t + 2})*\cos(t/\sqrt{t + 2})\).

Step 5 ：Final Answer: The derivative of the function \(q=\sin \left(\frac{t}{\sqrt{t+2}}\right)\) is \(\boxed{(-t/(2*(t + 2)^{3/2}) + 1/\sqrt{t + 2})*\cos(t/\sqrt{t + 2})}\)