Editing Logarithmic derivative (section)

==The multiplicative group==
Behind the use of the logarithmic derivative lie two basic facts about ''GL''<sub>1</sub>, that is, the multiplicative group of [[real number]]s or other [[field (mathematics)|field]]. The [[differential operator]]
<math display="block"> X\frac{d}{dX} </math>
is [[Invariant (mathematics)|invariant]] under dilation (replacing ''X'' by ''aX'' for ''a'' constant). And the [[differential form]] <math display="block">\frac{dx}{X}</math> is likewise invariant. For functions ''F'' into GL<sub>1</sub>, the formula
<math display="block">\frac{dF}{F}</math> is therefore a ''[[pullback (differential geometry)|pullback]]'' of the invariant form.{{Citation needed|date=August 2021}}