Find the derivative of the function. \[ y=x \sin \left(\frac{4}{x}\right) \] \[ y^{\prime}(x)= \] Need Help? Read It

Step 1 ：To find the derivative of the function, we can use the product rule and the chain rule. The product rule states that the derivative of two functions multiplied together 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 2 ：In this case, we have two functions: \(x\) and \(\sin\left(\frac{4}{x}\right)\). We will need to find the derivatives of these two functions and then apply the product rule.

Step 3 ：For the function \(x\), the derivative is simply 1.

Step 4 ：For the function \(\sin\left(\frac{4}{x}\right)\), we will need to use the chain rule. The outer function is \(\sin(x)\) and the inner function is \(\frac{4}{x}\). The derivative of \(\sin(x)\) is \(\cos(x)\) and the derivative of \(\frac{4}{x}\) is \(-\frac{4}{x^2}\).

Step 5 ：So, the derivative of \(\sin\left(\frac{4}{x}\right)\) is \(\cos\left(\frac{4}{x}\right) \cdot -\frac{4}{x^2}\).

Step 6 ：Now we can apply the product rule to find the derivative of the original function.

Step 7 ：The derivative of the function \(y=x \sin \left(\frac{4}{x}\right)\) is \(y^{\prime}(x)=\sin\left(\frac{4}{x}\right) - \frac{4\cos\left(\frac{4}{x}\right)}{x}\). This is the final answer.

Step 8 ：Final Answer: \(y^{\prime}(x)=\boxed{\sin\left(\frac{4}{x}\right) - \frac{4\cos\left(\frac{4}{x}\right)}{x}}\)