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The differential operator is invariant under dilation (replacing X by aX for a constant). And the differential form d x X {\displaystyle {\frac {dx}{X}}} is likewise invariant. For functions F into GL 1 , the formula d F F {\displaystyle {\frac {dF}{F}}} is therefore a pullback of the invariant form.
Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity. Integrals involving only logarithmic functions [ edit ]
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. ...
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, [1] () ′ = ′ ′ = () ′.
Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the coefficient of dx). Some authors and journals set the differential symbol d in roman type instead of italic: dx. The ISO/IEC 80000 scientific style guide recommends this style.
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
The p-forms with log poles along D form a subsheaf of the meromorphic p-forms on X, denoted Ω X p ( log D ) . {\displaystyle \Omega _{X}^{p}(\log D).} The name comes from the fact that in complex analysis , d ( log z ) = d z / z {\displaystyle d(\log z)=dz/z} ; here d z / z {\displaystyle dz/z} is a typical example of a 1-form on the ...