Is the Relationship Between Natural Log and Partial Derivatives in PDE Valid?

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SUMMARY

The relationship between the natural logarithm of a function \( k \) and its partial derivative with respect to \( P \) is confirmed as valid. Specifically, the equation \(\frac{\partial (ln(k))}{\partial P}=\frac{1}{k}\frac{\partial k}{\partial P}\) holds true, except at points where \( k \) equals zero or at branch cuts. This conclusion is derived from the application of the chain rule in calculus, affirming the interchangeability of the terms in the context of mineral physics.

PREREQUISITES
  • Understanding of partial derivatives in calculus
  • Familiarity with the chain rule in differentiation
  • Knowledge of natural logarithms and their properties
  • Basic concepts in mineral physics
NEXT STEPS
  • Study the application of the chain rule in multivariable calculus
  • Explore the properties of logarithmic functions in mathematical analysis
  • Investigate the implications of branch cuts in complex analysis
  • Research the role of partial derivatives in mineral physics applications
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Students and professionals in mathematics, physics, and engineering, particularly those focusing on partial differential equations and mineral physics.

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Is the following relationship true?:

\frac{\partial (ln(k))}{\partial P}=\frac{1}{k}\frac{\partial k}{\partial P}

I am getting both of these terms from a paper on mineral physics and they seem to use both terms interchangeably. If so, how are these related?
 
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log'(k)=1/k
So yes that is true except at zero or crossing branch cuts.
 
According to the chain rule, yes
 

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