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Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions.
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
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 () ′ = ( + ) ′ = () ′ + () ′.
Using that the logarithm of a product is the sum of the logarithms of the factors, the sum rule for derivatives gives immediately = = (). The last above expression of the derivative of a product is obtained by multiplying both members of this equation by the product of the f i . {\displaystyle f_{i}.}
Economic examples [ edit ] When campaigning for a second term in office, U.S. President Richard Nixon announced that the rate of increase of inflation was decreasing, which has been noted as "the first time a sitting president used the third derivative to advance his case for reelection."
Logarithmic differentiation; ... This is an example of the (ε, δ)-definition of ... The derivative of a composed function can be expressed using the chain rule: ...
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d. Derivations also use the log definitions x = b log b (x ...