Linearized error propagation for non-linear relationships

The approach developed in this section is of considerable practical importance for the assessment of errors on data obtained through a complex reducing procedure from raw measurements (e.g., optical and mass spectrometry), or on variables inferred through complex modeling. Given a relationship between a random variable X with mean px and variance rx2 and a dependent variable Y such as 

These equations can be subtracted from each order to give 

The useful equations (4.3.6) and (4.3.7), which are valid only for smooth monotonous functions, can be translated into relationships between the corresponding sample statistics 

Applying the equations (4.3.6) to (4.3.9) to highly non-linear functions, p is usually inappropriate. 

The derivative of the dependent variable eNd(0) relative to the independent variable (143Nd/144Nd)sample is 

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