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Logarithmic derivative

In mathematics, the logarithmic derivative of a differentiable function f (with f \neq 0) is defined as the quotient \frac{f'}{f}, where f' denotes the derivative of f.^[1] This expression equals the derivative of \log f wherever a branch of the logarithm exists, and it provides a way to study the relative rate of change of f without directly computing its full derivative.^[1] The concept arises naturally in calculus and extends to more advanced settings, including complex analysis, where it simplifies analysis of meromorphic functions with zeros, poles, or multiplicative structures.^[2] A key property of the logarithmic derivative is its additivity under multiplication: for differentiable functions f_1 and f_2 (with f_1 f_2 \neq 0), \frac{(f_1 f_2)'}{f_1 f_2} = \frac{f_1'}{f_1} + \frac{f_2'}{f_2}.^[1] In the complex domain, for a meromorphic function f, at a zero or pole of order m, \frac{f'}{f} has a simple pole with residue m (or -m for a pole).^[2] These features make it a powerful tool for decomposing complex functions into sums, facilitating computations in differentiation and integration.^[1] In complex analysis, the logarithmic derivative plays a central role in the argument principle, which states that for a meromorphic function f and a simple closed contour \gamma, the integral \frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)} \, dz equals the number of zeros minus the number of poles of f inside \gamma, counted with multiplicity.^[2] It is also essential in Nevanlinna theory, where the logarithmic derivative lemma provides growth estimates: for a meromorphic function f on the complex plane, the proximity function m(r, \frac{f'}{f}) satisfies m(r, \frac{f'}{f}) = O(\log T(r, f)) as r \to \infty, with T(r, f) the Nevanlinna characteristic.^[3] In number theory, the logarithmic derivative of the Riemann zeta function, \frac{\zeta'}{\zeta}(s), encodes information about the distribution of prime numbers via its partial fraction expansion over the non-trivial zeros, enabling explicit formulas that link primes to the zeros of \zeta.^[4]

Definition and Fundamentals

Definition

In mathematics, the logarithmic derivative is a concept that assumes familiarity with the basic notions of differentiation and the natural logarithm function. It applies to differentiable functions f defined on domains where f(x) \neq 0 to avoid singularities in the logarithm.^[5] Formally, for a differentiable function f(x) \neq 0, the logarithmic derivative is defined as the ratio \frac{f'(x)}{f(x)}, which is equivalent to the ordinary derivative of \ln |f(x)| for real-valued functions.^[6]^[5] In the complex domain, for a meromorphic function f(z), it is \frac{f'(z)}{f(z)}, corresponding to the derivative of a chosen branch of \log f(z) wherever defined, typically the principal branch where the argument of f(z) is restricted to -\pi < \operatorname{Arg}(f(z)) \leq \pi.^[7]^[8] The notation commonly used is \frac{f'}{f} or (\log f)', with the absolute value in the real case to ensure the logarithm is well-defined for both positive and negative values of f(x), and the principal branch specified in complex analysis to handle the multi-valued nature of the logarithm.^[6]^[5] This concept was introduced by Leonhard Euler in the 18th century, particularly in his 1755 work Institutiones calculi differentialis, to simplify the differentiation of exponential and logarithmic expressions.^[9]

Basic Properties

The logarithmic derivative of a function f, defined as \frac{f'}{f}, exhibits several fundamental algebraic properties that arise directly from the properties of the logarithm and the chain rule. Unlike the ordinary derivative, which is additive for sums, the logarithmic derivative is not generally additive in that form: \frac{(f + g)'}{f + g} \neq \frac{f'}{f} + \frac{g'}{g}.^[5] However, it is additive for products, reflecting the logarithm's property \log(fg) = \log f + \log g:

\frac{(fg)'}{fg} = \frac{f'}{f} + \frac{g'}{g}.

This property simplifies differentiation of multiplicative expressions.^[5]^[10] A corresponding adaptation of the quotient rule holds, based on \log(f/g) = \log f - \log g:

\frac{(f/g)'}{f/g} = \frac{f'}{f} - \frac{g'}{g}.

This follows from algebraic manipulation and is useful for ratios of functions.^[5]^[10] For compositions, the chain rule yields a multiplicative form: the logarithmic derivative of f(g(x)) is

\frac{d}{dx} \log f(g(x)) = \frac{f'(g(x))}{f(g(x))} \cdot g'(x).

This combines the relative change in f at g(x) with the ordinary derivative of g.^[5] Analytically, the logarithmic derivative \frac{f'}{f} represents the instantaneous relative rate of change of f, or the percentage growth per unit change in the independent variable; for instance, a value of 0.05 indicates a 5% increase per unit.^[10] This interpretation is particularly valuable in modeling growth processes where absolute and relative scales differ. Regarding uniqueness, if two differentiable functions f and h (with h > 0) satisfy \frac{f'}{f} = \frac{h'}{h} on an interval, then f = c h for some constant c > 0; this follows from integrating both sides to obtain \log f = \log h + k, or f = e^k h.^[11] In complex analysis, these properties extend to meromorphic functions, aiding analytic continuation.^[5]

Differentiation Techniques

Derivatives of Products and Quotients

The logarithmic derivative provides a streamlined approach to finding the derivatives of products of functions by leveraging the additivity of logarithms. Consider the product y = u(x) v(x), where u(x) > 0 and v(x) > 0 for the domain of interest. Taking the natural logarithm yields \ln y = \ln u + \ln v. Differentiating both sides with respect to x gives

\frac{1}{y} \frac{dy}{dx} = \frac{u'}{u} + \frac{v'}{v},

where primes denote ordinary derivatives. Multiplying through by y results in

\frac{dy}{dx} = y \left( \frac{u'}{u} + \frac{v'}{v} \right) = u v \left( \frac{u'}{u} + \frac{v'}{v} \right) = u' v + u v'.

This derivation recovers the standard product rule while expressing it in terms of logarithmic derivatives \frac{u'}{u} and \frac{v'}{v}, which can be particularly useful when u and v are themselves complex.^[12] For quotients, the same technique applies using the difference property of logarithms. Let y = \frac{u(x)}{v(x)}, assuming u(x) > 0 and v(x) > 0. Then \ln y = \ln u - \ln v, and differentiating produces

\frac{1}{y} \frac{dy}{dx} = \frac{u'}{u} - \frac{v'}{v}.

Multiplying by y yields

\frac{dy}{dx} = y \left( \frac{u'}{u} - \frac{v'}{v} \right) = \frac{u}{v} \left( \frac{u'}{u} - \frac{v'}{v} \right).

Expanding algebraically confirms the quotient rule:

\frac{dy}{dx} = \frac{u' v - u v'}{v^2},

as the right-hand side simplifies to \frac{u' v}{v^2} - \frac{u v'}{v^2}. This form highlights how the logarithmic derivative subtracts the relative rates of change for numerator and denominator.^[13] The primary advantage of this method lies in reducing the differentiation of functions with multiple multiplicative factors to a sum or difference of simpler logarithmic derivatives, avoiding repeated applications of the product or quotient rules. For instance, consider y = \frac{x^2 \sin x}{e^x}. Taking \ln y = 2 \ln x + \ln (\sin x) - x and differentiating gives

\frac{y'}{y} = \frac{2}{x} + \cot x - 1,

y' = y \left( \frac{2}{x} + \cot x - 1 \right) = \frac{x^2 \sin x}{e^x} \left( \frac{2}{x} + \cot x - 1 \right).

This avoids expanding the product x^2 \sin x and then applying the quotient rule directly with the exponential, streamlining the process for functions involving powers, trigonometrics, and exponentials.^[14]^[15] A key limitation is that the functions must be positive where the logarithm is applied, as \ln f is undefined for f \leq 0; in such cases, derivatives can be handled via limits as the argument approaches the boundary or by defining the function piecewise on intervals where it is positive.^[14]

Derivatives of Powers and Compositions

The logarithmic derivative provides a streamlined approach to finding the derivatives of power functions. Consider a function of the form y = [f(x)]^n, where n is a real constant and f(x) > 0. Taking the natural logarithm yields \ln y = n \ln f(x). Differentiating both sides with respect to x gives \frac{1}{y} y' = n \frac{f'(x)}{f(x)}, so y' = n [f(x)]^n \frac{f'(x)}{f(x)}. This is equivalent to the standard power rule y' = n [f(x)]^{n-1} f'(x), but expressed multiplicatively as y' = [f(x)]^n \cdot n \left( \frac{f'(x)}{f(x)} \right), highlighting the role of the logarithmic derivative \frac{f'(x)}{f(x)}.^[14]^[16] For exponential functions, the logarithmic derivative similarly simplifies computation. Let y = e^{g(x)}. Then \ln y = g(x), and differentiating both sides produces \frac{1}{y} y' = g'(x), yielding y' = e^{g(x)} g'(x). This can be rewritten as y' = y \cdot \left( \frac{d}{dx} \ln y \right) = e^{g(x)} \cdot g'(x), where \left( \frac{d}{dx} \ln e^u \right) = u' for u = g(x), underscoring the identity \frac{d}{dx} e^u = e^u u' through the logarithmic derivative.^[17] In compositions involving powers and exponentials, the logarithmic derivative facilitates handling higher-order derivatives, often via repeated application akin to the Leibniz rule for products, which extends to powers through the binomial theorem. For instance, the higher derivatives of e^{\ln f(x)} = f(x) recover f(x) itself. For functions with variable exponents, such as y = [f(x)]^{g(x)} where f(x) > 0, logarithmic differentiation yields \ln y = g(x) \ln f(x). Differentiating both sides gives \frac{1}{y} y' = g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)}, so y' = [f(x)]^{g(x)} \left[ g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)} \right]. This expression combines the logarithmic derivatives of f and the product structure of g \ln f, providing a general form for such compositions.^[14]

Differential Equations

Integrating Factors

In the context of first-order linear ordinary differential equations (ODEs), the standard form is given by y' + P(x)y = Q(x), where P(x) and Q(x) are functions of x. An integrating factor \mu(x) is a function that, when multiplied through the equation, transforms the left-hand side into an exact derivative. This factor is constructed as \mu(x) = e^{\int P(x) \, dx}, and its logarithmic derivative satisfies \frac{d}{dx} \log |\mu(x)| = \frac{\mu'(x)}{\mu(x)} = P(x).^[18]^[19] The derivation begins by multiplying the ODE by \mu(x), yielding \mu(x) y' + \mu(x) P(x) y = \mu(x) Q(x). For the left-hand side to equal \frac{d}{dx} [\mu(x) y] = \mu(x) y' + \mu'(x) y, the condition \mu'(x) = \mu(x) P(x) must hold. Dividing both sides by \mu(x) gives \frac{\mu'(x)}{\mu(x)} = P(x), which is precisely the logarithmic derivative of \mu(x). Integrating both sides produces \log |\mu(x)| = \int P(x) \, dx + C, so \mu(x) = e^{\int P(x) \, dx} (absorbing the constant into the exponential). This ensures the transformed equation is \frac{d}{dx} [\mu(x) y] = \mu(x) Q(x), which can then be integrated directly.^[18]^[11] To construct the integrating factor explicitly, compute the integral \int P(x) \, dx and exponentiate the result. For instance, consider the ODE y' + \frac{2}{x} y = x, where P(x) = \frac{2}{x}. Then \int P(x) \, dx = \int \frac{2}{x} \, dx = 2 \log |x| + C, so \mu(x) = e^{2 \log |x|} = x^2 (for x > 0, ignoring the constant). Multiplying through gives x^2 y' + 2x y = x^3, or \frac{d}{dx} (x^2 y) = x^3, which integrates to x^2 y = \frac{1}{4} x^4 + C. Another example is y' + y = e^x, with P(x) = 1, yielding \int 1 \, dx = x + C and \mu(x) = e^x. These computations highlight how the logarithmic derivative directly guides the selection of \mu(x) through antiderivative evaluation.^[19]^[11] This approach generalizes to first-order ODEs that are not initially exact, where the equation M(x,y) dx + N(x,y) dy = 0 fails \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}. The logarithmic derivative helps identify an integrating factor \mu(x) (depending only on x) if \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} is a function of x alone, say P(x), leading to \mu(x) = e^{\int P(x) \, dx} by matching the form to make the equation exact.^[18]

First-Order Linear ODEs

First-order linear ordinary differential equations (ODEs) take the standard form y' + P(x) y = Q(x), where P(x) and Q(x) are functions of the independent variable x. The logarithmic derivative plays a key role in constructing the integrating factor \mu(x) = \exp\left( \int P(x) \, dx \right), as the condition for \mu(x) derives from requiring \frac{\mu'(x)}{\mu(x)} = P(x), which is precisely the logarithmic derivative of \mu(x).^[20] Multiplying the ODE by \mu(x) transforms the left side into the exact derivative \frac{d}{dx} [\mu(x) y] = \mu(x) Q(x), allowing integration to yield the solution y(x) = \frac{1}{\mu(x)} \left[ \int \mu(x) Q(x) \, dx + C \right], where C is the constant of integration.^[21] The step-by-step process begins by identifying P(x) and Q(x) from the ODE. Compute \int P(x) \, dx (up to a constant, which does not affect \mu(x) since it exponentiates to a multiplicative constant absorbed elsewhere), then form \mu(x) = \exp\left( \int P(x) \, dx \right). Integrate \mu(x) Q(x) with respect to x, add C, and divide by \mu(x) to solve for y(x). For the homogeneous case y' + P(x) y = 0, the solution simplifies to y(x) = \frac{C}{\mu(x)}, as the integral term vanishes.^[21] In special cases with constant coefficients, such as P(x) = a (a constant), the integral \int P(x) \, dx = a x, so \mu(x) = e^{a x}, making the process explicit and straightforward.^[21] Bernoulli equations, a nonlinear extension of the form y' + P(x) y = Q(x) y^n with n \neq 0, 1, reduce to linear via the substitution v = y^{1-n}. Differentiating gives v' = (1-n) y^{-n} y', so \frac{y'}{y} = \frac{1}{1-n} \frac{v'}{v}, linking the original logarithmic derivative \frac{y'}{y} to that of v. Substituting yields a first-order linear ODE in v, solvable as above, then back-substitute for y = v^{1/(1-n)}.^[22] To verify a solution, differentiate y(x) explicitly and substitute into the original ODE, confirming it holds identically; this often involves checking the logarithmic derivative term \frac{y'}{y} matches the expected form from P(x) and Q(x).^[20]

Worked Example: Standard Nonhomogeneous Case

Consider the ODE y' + \frac{1}{x} y = x, for x > 0. Here, P(x) = \frac{1}{x} and Q(x) = x. Compute \int P(x) \, dx = \int \frac{1}{x} \, dx = \ln x, so \mu(x) = e^{\ln x} = x. Multiply through: x y' + y = x^2, or \frac{d}{dx} (x y) = x^2. Integrate: x y = \int x^2 \, dx = \frac{x^3}{3} + C. Thus, y(x) = \frac{x^2}{3} + \frac{C}{x}.^[21] Verification: Differentiate y = \frac{x^2}{3} + \frac{C}{x} to get y' = \frac{2x}{3} - \frac{C}{x^2}. Substitute: y' + \frac{1}{x} y = \left( \frac{2x}{3} - \frac{C}{x^2} \right) + \frac{1}{x} \left( \frac{x^2}{3} + \frac{C}{x} \right) = \frac{2x}{3} - \frac{C}{x^2} + \frac{x}{3} + \frac{C}{x^2} = x, which matches Q(x). Note \frac{y'}{y} = P(x) - \frac{Q(x)}{y} holds by construction.^[20]

Worked Example: Homogeneous Case

For y' + \frac{2}{x} y = 0, P(x) = \frac{2}{x}, Q(x) = 0. Then \int P(x) \, dx = 2 \ln x, \mu(x) = e^{2 \ln x} = x^2. The solution is y(x) = \frac{C}{x^2}. Verification: y' = -\frac{2C}{x^3}, so y' + \frac{2}{x} y = -\frac{2C}{x^3} + \frac{2}{x} \cdot \frac{C}{x^2} = 0.^[21]

Worked Example: Constant Coefficients

Solve y' - 3 y = e^{4x}. Here, P(x) = -3, \int P \, dx = -3x, \mu(x) = e^{-3x}. Multiply: e^{-3x} y' - 3 e^{-3x} y = e^{x}, or \frac{d}{dx} (e^{-3x} y) = e^{x}. Integrate: e^{-3x} y = \int e^{x} \, dx = e^{x} + C. Thus, y(x) = e^{4x} + C e^{3x}. Verification confirms substitution yields the ODE identically.^[21]

Worked Example: Bernoulli Equation

For y' + \frac{1}{x} y = \frac{1}{x} y^2, n=2, so $1-n = -1, v = y^{-1} = \frac{1}{y}. Then v' = - y^{-2} y', so y' = - y^2 v' = -\frac{1}{v^2} v'. Substitute: -\frac{1}{v^2} v' + \frac{1}{x} \frac{1}{v} = \frac{1}{x} \frac{1}{v^2}. Multiply by -v^2: v' - \frac{1}{x} v = -\frac{1}{x}. This is linear with P(x) = -\frac{1}{x}, Q(x) = -\frac{1}{x}. Compute \int P \, dx = -\ln x, \mu(x) = e^{-\ln x} = \frac{1}{x}. Multiply: \frac{1}{x} v' - \frac{1}{x^2} v = -\frac{1}{x^2}, or \frac{d}{dx} \left( \frac{v}{x} \right) = -\frac{1}{x^2}. Integrate: \frac{v}{x} = \int -\frac{1}{x^2} \, dx = \frac{1}{x} + C, so v = 1 + C x. Thus, y = \frac{1}{v} = \frac{1}{1 + C x}. The logarithmic link appears in \frac{v'}{v} = (1-n) \frac{y'}{y} = -\frac{y'}{y}. Verification by substitution confirms the original ODE.^[22]

Advanced Applications

Complex Analysis

In complex analysis, the logarithmic derivative of a holomorphic function f: U \to \mathbb{C}, where U is an open subset of \mathbb{C} and f(z) \neq 0, is defined as \frac{f'(z)}{f(z)}, which equals the derivative of \log f(z) using a suitable branch of the complex logarithm.^[23] The complex logarithm is multi-valued, so to ensure holomorphicity, one typically employs the principal branch \operatorname{Log} f(z) = \ln |f(z)| + i \operatorname{Arg} f(z) with \operatorname{Arg} f(z) \in (-\pi, \pi], defined on domains where f avoids the non-positive real axis; this yields a holomorphic antiderivative for \frac{f'}{f} locally away from branch cuts.^[24] The logarithmic derivative exhibits singularities precisely at the zeros and poles of f. If f has a zero of multiplicity m \geq 1 at z_0, then \frac{f'(z)}{f(z)} has a simple pole at z_0 with residue m.^[23] Conversely, if f has a pole of multiplicity m \geq 1 at z_0, then \frac{f'(z)}{f(z)} has a simple pole at z_0 with residue -m.^[23] These residues arise from the local Laurent series expansions of f near z_0, where near a zero, f(z) = (z - z_0)^m g(z) with g(z_0) \neq 0 and g holomorphic, leading to \frac{f'(z)}{f(z)} = \frac{m}{z - z_0} + \frac{g'(z)}{g(z)}.^[23] A key application is the argument principle, which connects the logarithmic derivative to the topology of meromorphic functions. For a meromorphic function f on a domain containing a simple closed positively oriented contour \gamma, with f holomorphic and non-zero on \gamma,

\frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)} \, dz = N - P,

where N is the number of zeros inside \gamma (counted with multiplicity) and P is the number of poles inside \gamma (counted with multiplicity).^[23] This follows from the residue theorem applied to \frac{f'}{f}, as the integral sums the residues at the interior zeros and poles, each contributing +m or -m respectively, while residues on \gamma vanish.^[23] The result equates to the winding number of the image curve f \circ \gamma around 0, providing a means to count zeros and poles via contour integrals. In the Weierstrass factorization theorem, the logarithmic derivative plays a central role in constructing infinite products for entire functions. For an entire function f with zeros \{a_k\} (counted with multiplicity), the Weierstrass form is f(z) = z^m e^{g(z)} \prod_k E_{p_k}(z/a_k), where E_p(u) = (1 - u) \exp(u + \cdots + u^p/p) are Weierstrass factors and g is entire; taking the logarithmic derivative yields

\frac{f'(z)}{f(z)} = \frac{m}{z} + g'(z) + \sum_k \frac{E_{p_k}'(z/a_k)/E_{p_k}(z/a_k)}{z/a_k} = \sum_k \frac{1}{z - a_k} + h(z),

where h is entire, reflecting the principal parts at the zeros plus a holomorphic correction.^[25] The Hadamard product refines this for functions of finite order \rho, setting p = \lfloor \rho \rfloor and g a polynomial of degree at most \rho, as in the example \sin(\pi z) = \pi z \prod_{k=1}^\infty (1 - z^2/k^2), whose logarithmic derivative is \pi \cot(\pi z) = 1/z + \sum_{k=1}^\infty (1/(z - k) + 1/(z + k)).^[25]

Multiplicative Structures

In abstract algebraic settings, the logarithmic derivative extends to multiplicative structures such as the group of units in a ring or field equipped with a derivation. For a commutative ring R with derivation \delta: R \to R, the multiplicative group G = R^\times of units admits a logarithmic derivative \delta(g) = g^{-1} \delta(g) for g \in G, which maps into the additive group underlying the module structure of R. This operation satisfies the homomorphism property \delta(gh) = \delta(g) + \delta(h) for g, h \in G, rendering it a derivation from the multiplicative group G to its associated Lie algebra, often the additive group of the base ring or field.^[26]^[27] In the specific case of matrix groups like \mathrm{GL}_n(K) over a differential field K with derivation \partial, the logarithmic derivative is given by Y \mapsto \partial(Y) Y^{-1} for Y \in \mathrm{GL}_n(K), yielding a crossed homomorphism into the Lie algebra \mathfrak{gl}_n(K). This construction is central to algebraic D-groups in differential Galois theory, where it characterizes the differential Galois group of linear equations \partial y = A y via the fundamental solution matrix Y satisfying \partial(Y) Y^{-1} = A. Surjectivity holds in differentially closed extensions, linking the image to the full Lie algebra.^[27] For formal power series over a field K of characteristic zero, consider the multiplicative group of units in K[] consisting of series f(x) = 1 + \sum_{n \geq 1} a_n x^n. The logarithmic derivative f'/f, where ' = d/dx, is itself a formal power series in K[], with leading term a_1 + 2 a_2 x + \cdots adjusted by higher-order interactions from the inverse $1/f = \sum_{k=0}^\infty (-1)^k (f-1)^k. This structure is pivotal in formal group theory, particularly for the exponential and logarithm formal groups, where the logarithm series L(x) satisfies L'(x) = 1 / f(x) for the group law's invariant differential f(x), facilitating isomorphisms between additive and multiplicative formal groups.^[28] In applications to Galois theory and differential Galois theory, the logarithmic derivative quantifies infinitesimal changes within multiplicative extensions, aiding the analysis of Picard-Vessiot extensions and solvability by quadratures. For instance, in solving the functional equation f' = f^2, the logarithmic derivative transforms it into a linear equation via u = 1/f, where \delta u = -u^2 + \delta(1/f), revealing the differential Galois group as a subgroup of \mathrm{PGL}_2, often the full projective general linear group for generic coefficients. This measures the "infinitesimal" symmetry breaking in the solution space.^[27]^[29] In p-adic fields, the logarithmic derivative connects to valuations through extensions of the arithmetic derivative to \mathbb{Q}_p, defined as \mathrm{ld}_K(x) = D_K(x)/x for x \in K^\times, where D_K preserves the product rule and relates to the p-adic valuation v_p. For units in p-adic completions, it interpolates logarithmic derivatives of power series associated to Lubin-Tate formal groups, with valuation properties ensuring continuity in locally analytic functions, such as v(f_{tp}) = -1/(p-1) under ramification constraints. This linkage facilitates p-adic interpolation in number theory, tying multiplicative structures to differentia in non-archimedean settings.^[30]^[31]

Illustrative Examples

Algebraic Functions

The logarithmic derivative of an algebraic function, defined as \frac{f'(x)}{f(x)}, simplifies the analysis of polynomials and rational functions by transforming multiplicative structures into additive ones via differentiation of the logarithm. For a general polynomial f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0, the derivative f'(x) = n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1 follows from the power rule, so the logarithmic derivative is \frac{f'(x)}{f(x)} = \frac{n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1}{x^n + a_{n-1} x^{n-1} + \cdots + a_0}. This ratio highlights the relative rate of change and is particularly useful in root-finding algorithms or stability analysis, where the numerator approximates the behavior near large x while the denominator captures global structure.^[32] A concrete example is the quadratic polynomial f(x) = x^2 + 1, where f'(x) = 2x and thus \frac{f'(x)}{f(x)} = \frac{2x}{x^2 + 1}. For rational functions, consider f(x) = \frac{x^2 + 1}{x - 1}; the derivative is f'(x) = \frac{2x(x-1) - (x^2 + 1)}{(x-1)^2} = \frac{x^2 - 2x - 1}{(x-1)^2}, so the logarithmic derivative simplifies to \frac{f'(x)}{f(x)} = \frac{x^2 - 2x - 1}{(x-1)(x^2 + 1)}. This form aids in partial fraction decomposition for integration or series expansion, revealing poles at the roots of the denominator.^[32] When expressed in factored form, f(x) = c \prod_{i=1}^n (x - r_i) for distinct roots r_i, the logarithmic derivative decomposes as \frac{f'(x)}{f(x)} = \sum_{i=1}^n \frac{1}{x - r_i}, a direct consequence of the product rule applied to \log f(x) = \log c + \sum \log(x - r_i). This partial fraction representation connects to the residue theorem in complex analysis, where each term \frac{1}{x - r_i} has residue 1 at r_i, facilitating zero counting via contour integrals. For roots with multiplicity m_j at r_j, the logarithmic derivative is exactly \sum_j \frac{m_j}{x - r_j}.^[32] Near a simple root r, the logarithmic derivative approximates \frac{1}{x - r}, dominating the behavior and indicating a pole of order 1, which underscores the function's sensitivity to perturbations around roots. For multiplicity, consider f(x) = (x^2 - 1)^2 = (x-1)^2 (x+1)^2; the derivative is f'(x) = 4x (x^2 - 1), so the logarithmic derivative is \frac{f'(x)}{f(x)} = \frac{4x}{x^2 - 1} = \frac{2}{x-1} + \frac{2}{x+1}, confirming the multiplicity contribution of order 2 at each root. This illustrates how multiplicities amplify the local pole strength in the logarithmic derivative.^[32] To compute the logarithmic derivative practically, apply the quotient rule directly to f(x) rather than differentiating \log f(x) first, as the latter requires evaluating the inverse hyperbolic tangent or similar for verification but yields the same result via the chain rule; this direct approach avoids unnecessary logarithmic evaluations for algebraic cases.^[32]

Transcendental Functions

The logarithmic derivative of the exponential function e^x is constantly 1, as its ordinary derivative equals the function itself. More generally, the logarithmic derivative of e^{g(x)} simplifies to g'(x), reflecting the chain rule applied to the exponential composition.^[5] For the natural logarithm \ln x (defined for x > 0), the ordinary derivative is $1/x, so the logarithmic derivative is \frac{1/x}{\ln x} = \frac{1}{x \ln x}. This form highlights the reciprocal relationship between the function and its growth rate relative to itself.^[33] In trigonometric functions, the logarithmic derivative of \sin x is \cot x = \frac{\cos x}{\sin x}, a result that appears prominently in infinite product representations and partial fraction expansions of the cotangent. Similarly, for \tan x, the ordinary derivative \sec^2 x yields the logarithmic derivative \frac{\sec^2 x}{\tan x} = \frac{1}{\sin x \cos x} = \frac{2}{\sin 2x}, which simplifies using double-angle identities and underscores connections to periodic structures.^[34] Hyperbolic functions exhibit analogous behavior due to their exponential definitions. The logarithmic derivative of \tanh x = \frac{\sinh x}{\cosh x} is \frac{\sech^2 x}{\tanh x} = \frac{1}{\sinh x \cosh x} = \frac{2}{\sinh 2x}, mirroring the trigonometric case but with hyperbolic identities. This equivalence arises from the relation \sinh x = -i \sin(ix) and \cosh x = \cos(ix). Series expansions of logarithmic derivatives for transcendental functions often involve special numbers. For instance, the logarithmic derivative of e^x - 1 is \frac{e^x}{e^x - 1}, whose Laurent series around x = 0 (with a simple pole) expands as $1 + \sum_{n=0}^\infty \frac{B_n}{n!} x^{n-1}, where B_n are the Bernoulli numbers (with the convention B_1 = -1/2). This connection links the expression to generating functions in analysis, such as those for sums and zeta values.^[35]

References

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