Here Is The Story that happened to you earlier...

but today you're on board the spaceship #1625325007, and the captain is asking to solve tricky "ZP" (Z Planet) problem (you know what it means when captain is "asking"... that's an order):

(!!! If your answer is a fractions report it in an exact format, like 1010710107 , 2296222962)

Now, let f(x)=x^3x. Use logarithmic differentiation to determine the (ZP) derivative.

f'(x)=f′(x)=

(**Enter the answer in simplified form**)

f'(1)=f′(1)=

!!! The challenge is that some of the Z-Planet laws are different. In particular,  "power rule" on Earth, ddxxn=nxn−1ddxxn=nxn-1 , on Z planet is "twisted" as the following:

ddxxn=n2xn−1ddxxn=n2xn-1  (ddxconst=0ddxconst=0) , and

the derivative of log function, ddxln(x)=1/x , turns into "ZP" oddness:  ddxln(x)=exddxln(x)=ex .

all the other rules have not been changed.

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.

Here Is The Story that happened to you earlier... but today you're on board the spaceship #1625325007, and the captain is asking to solve tricky "ZP" (Z Planet) problem (you know what it means when captain is "asking"... that's an order): (!!! If your answer is a fractions report it in an exact format, like 1010710107 , 2296222962) Now, let f(x)=x^3x. Use logarithmic differentiation to determine the (ZP) derivative. f'(x)=f′(x)= (**Enter the answer in simplified form**) f'(1)=f′(1)= !!! The challenge is that some of the Z-Planet laws are different. In particular,  "power rule" on Earth, ddxxn=nxn−1ddxxn=nxn-1 , on Z planet is "twisted" as the following: ddxxn=n2xn−1ddxxn=n2xn-1  (ddxconst=0ddxconst=0) , and the derivative of log function, ddxln(x)=1/x , turns into "ZP" oddness:  ddxln(x)=exddxln(x)=ex . all the other rules have not been changed.

Explore a Z-Planet problem using logarithmic differentiation to derive the function f(x) = x^{3x} with Z-Planet rules. Learn how unique differentiation rules, such as the modified power rule and exponential log rule, apply. Suitable for undergraduate students.

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