We are tasked with finding the derivative of $f(x) = x^{3x}$ under the unique Z-Planet differentiation laws. Let's break this problem down step-by-step, incorporating logarithmic differentiation and applying Z-Planet-specific rules.

Step 1: Express $f(x) = x^{3x}$

This expression is a power function where both the base and the exponent depend on $x$. To differentiate it, logarithmic differentiation is helpful.

Step 2: Take the natural logarithm of both sides

Taking the natural logarithm of both sides:

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

Using the logarithm property $\ln(a^b) = b \ln(a)$, we can rewrite this as:

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

Step 3: Differentiate both sides using the Z-Planet rules

Now, differentiate both sides with respect to $x$. On the left-hand side, apply the chain rule:

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

On the right-hand side, we differentiate $3x \ln(x)$ using the product rule $\frac{d}{dx}(uv) = u'v + uv'$, keeping in mind the Z-Planet rules for differentiation:

$\frac{d}{dx} (3x \ln(x)) = 3 \cdot 1 \cdot \ln(x) + 3x \cdot \frac{d}{dx} \ln(x)$

Using the Z-Planet rule for the derivative of the natural logarithm:

$\frac{d}{dx} \ln(x) = e^x$

So, the right-hand side becomes:

$3 \ln(x) + 3x e^x$

Step 4: Combine the results

We now have:

$\frac{1}{f(x)} \cdot f'(x) = 3 \ln(x) + 3x e^x$

Multiplying both sides by $f(x) = x^{3x}$, we obtain the derivative $f'(x)$:

$f'(x) = x^{3x} \left( 3 \ln(x) + 3x e^x \right)$

Step 5: Compute $f'(1)$

Now, let's substitute $x = 1$ into the expression for $f'(x)$:

$f(1) = 1^{3 \cdot 1} = 1$

For the derivative, we have:

$f'(1) = 1^{3 \cdot 1} \left( 3 \ln(1) + 3 \cdot 1 \cdot e^1 \right)$

Since $\ln(1) = 0$ and $e^1 = e$, this simplifies to:

$f'(1) = 1 \cdot \left( 0 + 3e \right) = 3e$

Final Answer:

• $f'(x) = x^{3x} \left( 3 \ln(x) + 3x e^x \right)$
• $f'(1) = 3e$

Let me know if you'd like more details or if anything is unclear!

Related Questions:

1. How does the Z-Planet power rule affect derivatives of polynomial functions?
2. What are some other Z-Planet-specific differentiation rules?
3. How would the derivative change if we used a different base for the exponent?
4. Can we generalize this method to functions like $f(x) = x^{nx}$?
5. How does logarithmic differentiation work on Earth compared to Z-Planet?

Tip:

Logarithmic differentiation is especially useful when both the base and the exponent of a function involve $x$, as it simplifies otherwise complex differentiation tasks.