Math Problem Statement
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.
Solution
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Math Problem Analysis
Mathematical Concepts
Logarithmic Differentiation
Exponential Functions
Product Rule
Formulas
Logarithmic differentiation: \( \ln(f(x)) \)
Z-Planet power rule: \( \frac{d}{dx} x^n = n^2 x^{n-1} \)
Z-Planet log derivative: \( \frac{d}{dx} \ln(x) = e^x \)
Theorems
Product Rule
Chain Rule
Suitable Grade Level
Undergraduate Level
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