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

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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:
$$
y^{\prime}(x) = x^{6} \sin(x) (7 + x \cos(x))
$$
$$ y^{\prime}(x) = x^{6} \sin(x) (7 + x \cos(x)) $$

Use logarithmic differentiation to find the derivative of the function.

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