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)=1xddxln(x)=1/x , turns into "ZP" oddness:  ddxln(x)=exddxln(x)=ex .

all the other rules have not been changed.

Let's solve for the derivative of \( f(x) = x^{3x} \) using logarithmic differentiation on Z-Planet.

### Step 1: Logarithmic Differentiation
Start by taking the natural logarithm of both sides:
\[
\ln f(x) = \ln (x^{3x}) = 3x \ln x
\]
Now differentiate both sides with respect to \( x \), using the Z-Planet differentiation rules.

### Step 2: Differentiate Using Z-Planet Rules
The Z-Planet rules are:
1. \( \frac{d}{dx} x^n = n^2 x^{n-1} \)
2. \( \frac{d}{dx} \ln(x) = e^x \)

Differentiate \( \ln f(x) \):
\[
\frac{d}{dx} \ln f(x) = \frac{d}{dx} (3x \ln x)
\]

Apply the product rule (which remains unchanged):
\[
\frac{d}{dx} (3x \ln x) = 3x \cdot \frac{d}{dx} (\ln x) + (\ln x) \cdot \frac{d}{dx} (3x)
\]

Now, calculate each derivative:
1. \( \frac{d}{dx} (\ln x) = e^x \)
2. \( \frac{d}{dx} (3x) = 3 \)

This gives:
\[
\frac{d}{dx} (3x \ln x) = 3x \cdot e^x + 3 \ln x
\]

### Step 3: Relate to \( f'(x) \)
Recall the relationship:
\[
\frac{1}{f(x)} f'(x) = 3x e^x + 3 \ln x
\]
Thus, solving for \( f'(x) \), we obtain:
\[
f'(x) = f(x) (3x e^x + 3 \ln x)
\]
Substitute \( f(x) = x^{3x} \):
\[
f'(x) = x^{3x} (3x e^x + 3 \ln x)
\]

### Step 4: Evaluate \( f'(1) \)
To find

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: Now, let f(x)=x^3x. Use logarithmic differentiation to determine the (ZP) derivative. f'(x)=? f'(1)=? The challenge is that some of the Z-Planet laws are different. In particular, ddxxn=nxn−1 on Earth, but on Z planet: ddxxn=n2xn−1 and the derivative of log function ddxln(x)=1/x on Earth, turns into: ddxln(x)=ex on Z planet.

Solve the Z-Planet problem using logarithmic differentiation for f(x)=x^3x. This step-by-step guide covers Z-Planet's unique differentiation rules, including twisted power and logarithmic rules. Suitable for Grades 10-12.

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