Search results
Results From The WOW.Com Content Network
It can also be useful when applied to functions raised to the power of variables or functions. 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."
In calculus, the power rule is used to differentiate functions of the form () =, whenever is a real number.Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule.
The chain rule can be used to derive some well-known differentiation rules. For example, the quotient rule is a consequence of the chain rule and the product rule. To see this, write the function f(x)/g(x) as the product f(x) · 1/g(x).
In economics, for example, consumer choice over a variety of goods, and producer choice over various inputs to use and outputs to produce, are modeled with multivariate calculus. Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus.