We are tasked with finding the derivative of $f(x) = x^{x^3}$ using logarithmic differentiation, while also keeping in mind the special Z-Planet (ZP) rules.

Step 1: Logarithmic Differentiation

Since $f(x) = x^{x^3}$, let's start by taking the natural logarithm of both sides of the equation to simplify the process.

$\ln(f(x)) = \ln(x^{x^3})$

Using the logarithmic identity $\ln(a^b) = b \ln(a)$, we get:

$\ln(f(x)) = x^3 \ln(x)$

Now, differentiate both sides with respect to $x$. Since we are differentiating logarithmic functions, we need to apply the Z-Planet rules.

Step 2: Differentiating the left-hand side

The derivative of $\ln(f(x))$ using the chain rule gives:

$\frac{d}{dx} \ln(f(x)) = \frac{1}{f(x)} f'(x)$

Step 3: Differentiating the right-hand side

We need to apply the product rule to $x^3 \ln(x)$.

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