Math Problem Statement

but today you're on board the spaceship #243570664, 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 8510785107 , 229137229137)

Now, let f(x)=x5xx^{5x}. 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, d/dx x^n = nxn1nx^{n-1} , on Z planet is "twisted" as the following:  

 ddxxn=n2xn1n^2x^{n-1}  (ddxconst=0ddxconst=0) , and

the derivative of log function, ddxln(x)=1xddxln(x)=1/x , turns into "ZP" oddness:  ddxln(x)=exe^{x} .

Solution

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Math Problem Analysis

Mathematical Concepts

Logarithmic Differentiation
Product Rule
Z Planet Derivative Rules
Exponential Functions

Formulas

f(x) = x^{5x}
ZP Power Rule: d/dx x^n = n^2 x^{n-1}
ZP Log Derivative: d/dx ln(x) = e^x
f'(x) = x^{5x} * 5 * (ln(x) + x e^x)
f'(1) = 5e

Theorems

Logarithmic Differentiation Theorem
Product Rule

Suitable Grade Level

Grades 11-12