HOME

TheInfoList



OR:

In
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function , (\ln f)' = \frac \quad \implies \quad f' = f \cdot (\ln f)'. The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
as well as properties of
logarithms In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
(in particular, the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, or the logarithm to the base '' e'') to transform products into sums and divisions into subtractions. The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero.


Overview

The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated. These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws are \ln(ab) = \ln(a) + \ln(b), \qquad \ln\left(\frac\right) = \ln(a) - \ln(b), \qquad \ln(a^n) = n\ln(a).


Higher order derivatives

Using Faà di Bruno's formula, the n-th order logarithmic derivative is, \frac \ln f(x) = \sum_ \frac \cdot \frac \cdot \prod_^n \left(\frac\right)^. Using this, the first four derivatives are, \begin \frac \ln f(x) &= \frac - \left(\frac \right)^2 \\ ex \frac \ln f(x) &= \frac - 3 \frac + 2 \left(\frac \right)^3 \\ ex \frac \ln f(x) &= \frac - 4 \frac - 3 \left(\frac\right)^2 + 12 \frac - 6 \left(\frac \right)^4 \end


Applications


Products

A
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
is applied to a product of two functions f(x) = g(x) h(x) to transform the product into a sum \ln(f(x))=\ln(g(x)h(x)) = \ln(g(x)) + \ln(h(x)). Differentiating by applying the chain and the sum rules yields \frac = \frac + \frac, and, after rearranging, yields f'(x) = f(x)\times \left\ = g(x) h(x) \times \left\ = g'(x) h(x) + g(x) h'(x), which is the product rule for derivatives.


Quotients

A
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
is applied to a quotient of two functions f(x) = \frac to transform the division into a subtraction \ln(f(x)) = \ln\left(\frac\right) = \ln(g(x)) - \ln(h(x)) Differentiating by applying the chain and the sum rules yields \frac = \frac - \frac, and, after rearranging, yields f'(x) = f(x) \times \left\ = \frac \times \left\ = \frac, which is the
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function (mathematics), function that is the ratio of two differentiable functions. Let h(x)=\frac, where both and are differentiable and g(x)\neq 0. The quotient rule sta ...
for derivatives.


Functional exponents

For a function of the form f(x) = g(x)^ the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
transforms the exponentiation into a product \ln(f(x)) = \ln\left(g(x)^\right) = h(x) \ln(g(x)) Differentiating by applying the chain and the product rules yields \frac = h'(x) \ln(g(x)) + h(x) \frac, and, after rearranging, yields f'(x) = f(x)\times \left\ = g(x)^ \times \left\. The same result can be obtained by rewriting ''f'' in terms of
exp Exp or EXP may stand for: * Exponential function, in mathematics * Expiry date of organic compounds like food or medicines * Experience point An experience point (often abbreviated as exp or XP) is a unit of measurement used in some tabletop r ...
and applying the chain rule.


General case

Using capital pi notation, let f(x) = \prod_i (f_i(x))^ be a finite product of functions with functional exponents. The application of natural logarithms results in (with capital sigma notation) \ln (f(x)) = \sum_i\alpha_i(x) \cdot \ln(f_i(x)), and after differentiation, \frac = \sum_i \left alpha_i'(x)\cdot \ln(f_i(x)) + \alpha_i(x) \cdot \frac\right Rearrange to get the derivative of the original function, f'(x) = \overbrace^ \times\overbrace^.


See also

* * * * * *


Notes

{{Calculus topics Differential calculus Logarithms