Use logarithmic differentiation to find the derivative of the function. \[ y=x^{7} \sin (x) \] \( y^{\prime}(x)=\square \)

To find the derivative of the function \( y = x^{7} \sin(x) \) using logarithmic differentiation, we will follow these steps: 1. **Take the natural logarithm of both sides:** \[ \ln(y) = \ln(x^{7} \sin(x)) \] 2. **Use the properties of logarithms to simplify the right side:** \[ \ln(y) = \ln(x^{7}) + \ln(\sin(x)) = 7 \ln(x) + \ln(\sin(x)) \] 3. **Differentiate both sides with respect to \( x \):** Using implicit differentiation on the left side and the chain rule on the right side: \[ \frac{1}{y} \frac{dy}{dx} = \frac{7}{x} + \frac{\cos(x)}{\sin(x)} = \frac{7}{x} + \cot(x) \] 4. **Multiply both sides by \( y \) to solve for \( \frac{dy}{dx} \):** \[ \frac{dy}{dx} = y \left( \frac{7}{x} + \cot(x) \right) \] 5. **Substitute back \( y = x^{7} \sin(x) \):** \[ \frac{dy}{dx} = x^{7} \sin(x) \left( \frac{7}{x} + \cot(x) \right) \] Now, we can simplify this expression: \[ \frac{dy}{dx} = x^{7} \sin(x) \left( \frac{7}{x} + \frac{\cos(x)}{\sin(x)} \right) = x^{7} \sin(x) \left( \frac{7 + x \cos(x)}{x} \right) \] Thus, the derivative \( y'(x) \) is: \[ y'(x) = x^{6} \sin(x) (7 + x \cos(x)) \] So, the final answer is: \[ y^{\prime}(x) = x^{6} \sin(x) (7 + x \cos(x)) \]