12. [-/1 Points] DETAILS MY NOTES SCALCET9 3.6.04 Use logarithmic differentiation to find the derivative of the function. \( y=x^{8 x} \) \( y^{\prime}(x)=\square \)

To find the derivative of the function \( y = x^{8x} \) using logarithmic differentiation, we will follow these steps: 1. **Take the natural logarithm of both sides**: \[ \ln(y) = \ln(x^{8x}) \] 2. **Use the properties of logarithms** to simplify the right side: \[ \ln(y) = 8x \ln(x) \] 3. **Differentiate both sides** with respect to \( x \). Remember to use the chain rule on the left side: \[ \frac{1}{y} \frac{dy}{dx} = 8 \ln(x) + 8x \cdot \frac{1}{x} \] Simplifying the right side gives: \[ \frac{1}{y} \frac{dy}{dx} = 8 \ln(x) + 8 \] 4. **Multiply both sides by \( y \)** to solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = y \left( 8 \ln(x) + 8 \right) \] 5. **Substitute back for \( y \)**: \[ \frac{dy}{dx} = x^{8x} \left( 8 \ln(x) + 8 \right) \] Now, we can express the final answer for the derivative \( y' \): \[ y' = x^{8x} (8 \ln(x) + 8) \] Thus, the derivative of the function \( y = x^{8x} \) is: \[ y' = x^{8x} (8 \ln(x) + 8) \]