To find the derivative \( \frac{d}{dx} \) of the function \( y = 8x^{\ln(x)} \), we will use **logarithmic differentiation** since the exponent \( \ln(x) \) is itself a function of \( x \). Let’s proceed step-by-step:

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### Step 1: Take the natural logarithm of both sides
Given:  
\[
y = 8x^{\ln(x)}
\]
Taking the natural logarithm of both sides:

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

Using the logarithm property \( \ln(ab) = \ln(a) + \ln(b) \):

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

Now apply \( \ln(a^b) = b \ln(a) \):

\[
\ln(y) = \ln(8) + \ln(x) \cdot \ln(x)
\]

This simplifies to:

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

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### Step 2: Differentiate both sides with respect to \( x \)
Using implicit differentiation on both sides:

\[
\frac{d}{dx} \left( \ln(y) \right) = \frac{d}{dx} \left( \ln(8) + (\ln(x))^2 \right)
\]

Since \( \frac{d}{dx}(\ln(y)) = \frac{1}{y} \cdot \frac{dy}{dx} \), and \( \ln(8) \) is a constant, we get:

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

Since \( \frac{d}{dx}(\ln(x)) = \frac{1}{x} \), the equation becomes:

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

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### Step 3: Solve for \( \frac{dy}{dx} \)
Multiplying both sides by \( y = 8x^{\ln(x)} \):

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

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### Step 4: Final Simplification
\[
\boxed{\frac{dy}{dx} = \frac{16 \ln(x) \cdot x^{\ln(x)}}{x}}
\]

Or equivalently:

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

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Let me know if you need further clarification or more details!

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Here are five related questions to explore:
1. What is logarithmic differentiation and when is it useful?
2. How do you apply implicit differentiation in calculus?
3. Can you explain why \( \frac{d}{dx}(\ln(x)) = \frac{1}{x} \)?
4. How does the derivative of a power function \( x^n \) change when the exponent is itself a function?
5. How would the derivative change if the function were \( 8x^{x} \) instead of \( 8x^{\ln(x)} \)?  

**Tip:** Logarithmic differentiation is particularly useful for differentiating functions where both the base and the exponent are functions of \( x \).

Learn how to differentiate the function 8x^(ln(x)) with respect to x. This solution applies logarithmic differentiation and explores advanced calculus techniques. Ideal for high school students in Grades 11-12 studying calculus.

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