Logarithmic differentiation explained

In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function,^[1] $$(\backslash ln\; f)\text{'}\; =\; \backslash frac\; \backslash quad\; \backslash implies\; \backslash quad\; f\text{'}\; =\; f\; \backslash cdot\; (\backslash ln\; f)\text{'}.$$

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 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.^{[2] [3]} 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.^[4] 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$$\backslash ln(ab)\; =\; \backslash ln(a)\; +\; \backslash ln(b),\; \backslash qquad\backslash ln\backslash left(\backslash frac\backslash right)\; =\; \backslash ln(a)\; -\; \backslash ln(b),\; \backslash qquad\backslash ln(a^n)\; =\; n\backslash ln(a).$$

Higher order derivatives

Using Faà di Bruno's formula, the n-th order logarithmic derivative is,$$\backslash frac\; \backslash ln\; f(x)=\; \backslash sum\_\; \backslash frac\; \backslash cdot\backslash frac\; \backslash cdot\backslash prod\_^n\; \backslash left(\backslash frac\backslash right)^.$$Using this, the first four derivatives are,

Applications

Products

See main article: Product rule. A natural logarithm is applied to a product of two functions$$f(x)\; =\; g(x)\; h(x)$$to transform the product into a sum$$\backslash ln(f(x))=\backslash ln(g(x)h(x))\; =\; \backslash ln(g(x))\; +\; \backslash ln(h(x)).$$Differentiating by applying the chain and the sum rules yields$$\backslash frac\; =\; \backslash frac\; +\; \backslash frac,$$and, after rearranging, yields^[5] $$f\text{'}(x)\; =\; f(x)\backslash times\; \backslash left\backslash \; =g(x)\; h(x)\; \backslash times\; \backslash left\backslash \; =\; g\text{'}(x)\; h(x)\; +\; g(x)\; h\text{'}(x),$$which is the product rule for derivatives.

Quotients

See main article: Quotient rule. A natural logarithm is applied to a quotient of two functions$$f(x)\; =\; \backslash frac$$to transform the division into a subtraction$$\backslash ln(f(x))\; =\; \backslash ln\backslash left(\backslash frac\backslash right)\; =\; \backslash ln(g(x))\; -\; \backslash ln(h(x))$$Differentiating by applying the chain and the sum rules yields$$\backslash frac\; =\; \backslash frac\; -\; \backslash frac,$$and, after rearranging, yields$$f\text{'}(x)\; =\; f(x)\; \backslash times\; \backslash left\backslash \; =\backslash frac\; \backslash times\; \backslash left\backslash \; =\; \backslash frac,$$

which is the quotient rule for derivatives.

Functional exponents

For a function of the form$$f(x)\; =\; g(x)^$$the natural logarithm transforms the exponentiation into a product$$\backslash ln(f(x))\; =\; \backslash ln\backslash left(g(x)^\backslash right)\; =\; h(x)\; \backslash ln(g(x))$$Differentiating by applying the chain and the product rules yields$$\backslash frac\; =\; h\text{'}(x)\; \backslash ln(g(x))\; +\; h(x)\; \backslash frac,$$and, after rearranging, yields$$f\text{'}(x)\; =\; f(x)\backslash times\; \backslash left\backslash \; =g(x)^\; \backslash times\; \backslash left\backslash .$$The same result can be obtained by rewriting f in terms of exp and applying the chain rule.

General case

Using capital pi notation, let$$f(x)\; =\; \backslash prod\_i\; (f\_i(x))^$$be a finite product of functions with functional exponents.

The application of natural logarithms results in (with capital sigma notation)$$\backslash ln\; (f(x))\; =\; \backslash sum\_i\backslash alpha\_i(x)\; \backslash cdot\; \backslash ln(f\_i(x)),$$and after differentiation,$$\backslash frac\; =\; \backslash sum\_i\; \backslash left[\backslash alpha\_i\text{'}(x)\backslash cdot\; \backslash ln(f\_i(x))\; +\; \backslash alpha\_i(x)\; \backslash cdot\; \backslash frac\{f\_i\text{'}(x)\}\{f\_i(x)\}\backslash right].$$Rearrange to get the derivative of the original function,$$f\text{'}(x)\; =\; \backslash overbrace^\; \backslash times\backslash overbrace^.$$

Notes and References

1. Book: Krantz, Steven G.. Calculus demystified. 170. McGraw-Hill Professional. 2003 . 0-07-139308-0.
2. Book: Golden Differential Calculus . 282. N.P. Bali. Firewall Media . 2005 . 81-7008-152-1.
3. Book: Bird, John. Higher Engineering Mathematics . 324 . Newnes . 2006 . 0-7506-8152-7.
4. Book: Blank, Brian E.. Calculus, single variable . 457. Springer. 2006. 1-931914-59-1.
5. Book: Williamson, Benjamin. An Elementary Treatise on the Differential Calculus . BiblioBazaar, LLC . 2008 . 25–26 . 978-0-559-47577-1.