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Show that T * ([tex]\partial[/tex]/ [tex]\partial[/tex]T) = (-1/T) * ([tex]\partial[/tex]/ [tex]\partial[/tex](1/T))

["[tex]\partial[/tex]/[tex]\partial[/tex]T" is the operator that takes the partial derivative of something with respect to T]

Showing that this is true is a little tricky. For example, we can define F = 1/T. Then ([tex]\partial[/tex]F/ [tex]\partial[/tex]T) = -1/T^2 and ([tex]\partial[/tex]F/ [tex]\partial[/tex]F) = 1. So we can write

([tex]\partial[/tex]F/ [tex]\partial[/tex]T) = (-1/T^2) ([tex]\partial[/tex]F/ [tex]\partial[/tex]F).

[In the next step he drops the F, so it's now an operator for an arbitrary function, but still with respect to F… Is this really okay?]

([tex]\partial[/tex]/ [tex]\partial[/tex]T) = (-1/T^2)([tex]\partial[/tex]/ [tex]\partial[/tex]F)

= (-1/T^2) ([tex]\partial[/tex]/ [tex]\partial[/tex](1/T)).

Multiplying by T, T([tex]\partial[/tex]/ [tex]\partial[/tex]T) = (-1/T)([tex]\partial[/tex]/ [tex]\partial[/tex](1/T)) and we're done.