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Concept
Logarithmic derivative
Summary
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula
where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f.
When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule:
Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have
So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get
Thus, it is true for any function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).
A corollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.
More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:
just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.
Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:
just as the logarithm of a power is the product of the exponent and the logarithm of the base.
In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.
Official source
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