To find the derivative \(\frac{dy}{dx}\) of the function \(y = 8x^{\ln(x)}\), we will apply logarithmic differentiation since the exponent \(\ln(x)\) depends on \(x\).

1. **Rewrite the function in terms of logarithms:**

   \[
   y = 8x^{\ln(x)}
   \]

   Take the natural logarithm of both sides to simplify:

   \[
   \ln(y) = \ln(8x^{\ln(x)}) = \ln(8) + \ln(x^{\ln(x)})
   \]

   Using the power rule for logarithms, \(\ln(x^{\ln(x)}) = \ln(x) \cdot \ln(x)\):

   \[
   \ln(y) = \ln(8) + (\ln(x))^2
   \]

2. **Differentiate both sides with respect to \(x\):**

   Using implicit differentiation:

   \[
   \frac{1}{y} \frac{dy}{dx} = 0 + 2 \ln(x) \cdot \frac{1}{x}
   \]

   Simplifying, we get:

   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{2 \ln(x)}{x}
   \]

3. **Solve for \(\frac{dy}{dx}\):**

   Multiply both sides by \(y\):

   \[
   \frac{dy}{dx} = y \cdot \frac{2 \ln(x)}{x}
   \]

   Substitute back \(y = 8x^{\ln(x)}\):

   \[
   \frac{dy}{dx} = 8x^{\ln(x)} \cdot \frac{2 \ln(x)}{x}
   \]

   Therefore, the derivative is:

   \[
   \frac{dy}{dx} = \frac{16x^{\ln(x) - 1} \ln(x)}{}
   \]

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Explore the solution for finding dy/dx of y = 8x^(ln(x)) with logarithmic differentiation. This calculus problem includes properties of logarithms, exponentials, and implicit differentiation, providing a clear step-by-step solution. Ideal for college-level calculus students.

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