Derivative of principal value (1/x)

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The principal value of 1/x is recognized as a generalized function or distribution. The discussion centers around finding its derivative, with a suggestion that it may be the principal value of -1/x^2. However, there is a debate about whether an explicit formula can be derived and then differentiated directly. The complexity of generalized functions complicates the differentiation process. Ultimately, the conversation highlights the challenges in handling derivatives of principal value distributions.
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The principal value of 1/x is a generalized function/distribution.

What is the derivative of it?

Is it the principal value of -1/x^2 ?
 
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Well, can't one write out an explicit formula and then just differentiate it?
 

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