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)=x3xf(x)=x3x. 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)=1x , 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
Derivatives
Exponential Functions
Product Rule
Formulas
f(x) = x^{x^3}
d/dx x^n = n^2 x^{n-1} (Z-Planet rule)
d/dx ln(x) = e^x (Z-Planet rule)
Product Rule: d(uv)/dx = u'v + uv'
Theorems
Logarithmic Differentiation Theorem
Product Rule
Exponential Derivative Rules
Suitable Grade Level
Undergraduate
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